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ELEMENTS 


OF 


GEOMETEY 


BY 


ELI    T.   TAPPAN,   LL.D., 

r-ROFESSOR  OF    POLtTICAL  SCIENCE  IN  KENTON   COLLEGE,   FORMERLY  PROFESSOR  OF 
MATHEMATICS. 


NEW   YORK: 
D.    APPLETON"   AND    COMPANY 

1,  3,  AND  5  BOND  STEEET. 
1885. 


T3 


COPTEIGHT,   1884, 

FY  D.  APPLETON  AND  COMPANY. 


<ll5!iF&0s> 


This  book  owes  mucli  to  criticisms,  made  by  many  of  my 
f ellow-teacbers,  upon  my  Treatise  on  Plane  and  Solid  Geome- 
try. To  my  pupils  also  I  am  greatly  indebted;  their  ques- 
tions and  remarks  have  indicated  many  difficulties  to  be 
smoothed  and  errors  to  be  avoided  in  an  elementary  work. 

It  has  been  my  aim — ^to  state  correctly  the  first  principles; 
of  the  science;   upon  these  premises  to  demonstrate  rigor- 
ously and  in  good  English  the  whole  doctrine  of  Elementary 
Geometry;  and  to  arrange  the  matter  so  as  to  proceed  with 
easy  gradation  from  the  simple  to  the  complex. 

Some  teachers  hold  that  the  study  of  geometry  is  valuable 
only  as  a  kind  of  intellectual  gymnastics.  Mr.  Todhunter  says 
that  the  admirers  of  Euclid  value  it  "more  for  the  process 
of  reasoning  involved  than  for  the  results  obtained";  and 
Mr.  De  Morgan  sums  up  the  educational  value  of  geometry 
in  the  habit  of  "tracing  necessary  consequences  from  given 
premises  by  elementary  logical  steps."  With  such  views, 
much  of  the  best  use  of  the  study  is  neglected.  It  is  as  im- 
portant to  begin  with  truth  as  to  argue  logically.  The  cer- 
tain knowledge  of  the  notions  we  begin  with  is  as  useful  as 
the  process  of  inference.     In  volumes  of  pretentious  and  false 

184009 


iv  PREFACE, 

theories,  almost  the  only  logical  errors  are  the  assumptions 
made  in  a  few  sentences. 

It  is  a  great  misfortune  to  a  youth  to  believe  that  the 
truth  of  premises  is  immaterial,  or  that  the  process  of  de- 
monstration is  the  paramount  object.  Injury  to  love  of  truth 
is  more  fatal  than  ignorance  of  logic.  No  science  can  grati- 
fy the  desire  for  exact  knowledge  better  than  this,  where 
the  truth  is  unmixed  with  errors  of  observation. 

In  England,  the  text-book  on  geometry  in  most  common 
use  has  been  Eobert  Simson's  version  of  the  Elements  of 
Euchd,  published  little  more  than  a  hundred  years  ago,  but 
written  in  a  style  that  was  even  then  growing  obsolete.  A 
quaint  language,  not  used  by  good  writers  on  any  other  science, 
has  been  the  custom  in  English  works  on  geometry.  Proba- 
bly I  have  not  been  able  to  avoid  this  entirely.  A  prime 
object  of  this  study  is  to  train  the  student  in  rigorous  argu- 
ment, but  the  training  is  of  more  value  when  the  argument 
is  expressed  in  the  common  language. 

The  undue  importance  allowed  to  syllogistic  inference  has 
shut  the  eyes  of  many  geometers  to  the  want  of  logical  order 
in  the  arrangement  of  topics.  EucHd's  first  proposition  treats 
of  a  triangle.  The  order  pursued  by  Legendre  is  somewhat 
more  gradual.  The  example  of  these  illustrious  authors  has 
induced  many  of  their  students  to  think  that  a  gradual  de- 
velopment of  the  subject  is  not  consistent  with  rigorous  de- 
monstration. Admitting  the  necessity  of  retaining  this  rigor, 
I  have  tried  to  show  that  it  is  aided  and  not  hindered  by  an 
arrangement  of  the  topics  in  the  order  of  their  increasing 
complexity,  beginning  with  straight  lines. 

In  the  earlier  chapters  I  have  usually  given  all  the  steps 
of  demonstration,  the  general  truth,  the  particular,  the  con- 


PREFACE.  Y 

struction,  the  argument,  and  the  conclusion,  according  to  the 
ancient  method.  In  the  last  two  or  three  chapters  more 
brevity  is  used.  Graduation  has  been  observed  also  in  the 
exercises;  hints  are  freely  given  in  the  beginning,  but  the 
student,  as  his  knowledge  increases,  is  supposed  to  have  ac- 
quired skill  by  practice. 

The  reasons  that  have  governed  my  work,  in  some  points 
about  which  geometers  differ,  are  presented  in  the  I^otes  more 
fully  than  would  be  suitable  in  the  limits  of  a  preface. 

Eli  T.  Tappan. 

Gambier,  Ohio,  October,  1884. 


CONTENTS. 


PART  FIRST.— ELEMENTARY. 

PAGE 

CHAPTER  L— Preliminary  .  .  .  .  .  .  .1 

CHAPTER  II.— The  Subject  Stated       .....  4 

PART  SECOND.— PLANE  GEOMETRY. 

CHAPTER  IIL— Straight  Lines        .            .            .            .            ,  .14 

CHAPTER  IV.— Circumferences             .            .            .            .            .  38 

Scholium  on  Geometrical  Reasoning       .            .  .57 

Problems  in  Drawing              ....  69 

CHAPTER  v.— Triangles      .            .            .            .            .            .  .     6Y 

CHAPTER  VI.— Quadrilaterals             .....  101 

CHAPTER  VII.— Polygons 123 

CHAPTER  VIII.— Circles 143 

PART  THIRD.— GEOMETRY  OF  SPACE. 

CHAPTER  IX.— Straight  Lines  and  Planes     .  .  .  .156 

CHAPTER  X.— PoLYEDRONS    .  .  .  .  .  .  .185 

CHAPTER  XL— Solids  of  Revolution  .  .  .  .  .212 

NOTES  .  . 243 

INDEX      .  .  . 249 


NOTICE 


In  this  work,  reference  to  an  article  in  tlie  same  chapter 
is  made  thus  (5),  that  is,  Article  5  of  the  chapter  in  which  the 
reference  is  given.  Reference  to  an  article  in  a  previous 
chapter,  thus  (Y,  6),  that  is.  Article  6,  Chapter  Y.  Eeference 
is  made  to  a  corollary  by  adding  the  number  in  small  capitals, 
thus  (5,  ii)  or  (Y,  6,  ii),  that  is.  Corollary  II  of  that  article. 
Reference  to  a  problem  in  drawing,  or  any  other  subdivision 
of  an  article,  is  made  in  the  same  way  as  to  a  corollary.  When 
there  is  only  one  corollary  in  an  article,  it  is  referred  to  by 
the  letter  c. 

In  the  exercises,  the  added  remark  or  reference  [in  brack- 
ets] is  given  as  a  hint  to  the  student. 


ELEMENTS  OF  GEOMETRY. 


CHAPTER  I. 
PRELIMINARY. 

Article  1.— The  principles  of  Geometry  are  applied  when- 
ever the  size,  shape,  or  position  of  anything  is  investigated. 
This  science  establishes  the  laws  upon  which  all  measurements 
are  made.  It  is  the  basis  of  the  sciences  of  Mechanics  and 
Astronomy.  Without  Geometry,  men  could  not  build  machines, 
survey  the  earth,  or  navigate  the  ocean. 

In  addition  to  these  uses  of  the  science,  Geometry  is  taught 
for  the  purpose  of  intellectual  training.  The  student  cultivates 
precision  of  language,  by  the  use  of  precise  terms  ;  his  reason- 
ing power,  in  the  various  analyses  and  demonstrations  ;  his 
imagination,  in  conceiving  and  holding  in  his  mind  the  com- 
binations of  lines  and  surfaces  ;  and  his  inventive  faculty,  in 
making  new  solutions  and  demonstrations. 

The  definitions  and  principles  in  this  chapter  are  not  peculiar 
to  Geometry,  but  are  much  used  in  the  study,  and  are  placed 
here  for  convenience  of  reference. 

liOgical  Terms. 

3,  Propositions  in  geometry  are  either  theoretical  or  practi- 
cal. Theoretical  propositions  declare  that  a  certain  property 
belongs  or  does  not  belong  to  a  certain  object ;  practical 
propositions  declare  that  something  can  be. 

Some  propositions  are  so  simple  and  evident  that  they  can 
not  be  made  more  so  by  any  course  of  reasoning.  They  are 
therefore   called  indemonstrable.      They  are  also   called  self- 


2  ELEMENTS   OF  GEOMETRY.  [Chap.  I. 

evident,  because  every  person  that  apprehends  the  meaning  of 
such  a  proposition  necessarily  admits  its  truth. 

A  Postulate  is  a  self-evident  practical  proposition. 

A  Problem  is  a  practical  proposition  that  can  be  proved. 

An  Axiom  is  a  self-evident  theoretical  proposition. 

A  Theorem  is  a  theoretical  proposition  that  can  be  de- 
monstrated. 

A  Corollary  is  a  proposition  that  follows  from  previous 
principles  without  further  reasoning,  or  the  demonstration  of 
which  is  brief  and  simple. 

Converse  propositions  are  two  propositions,  such  that  the 
subject  of  each  is  the  predicate  of  the  other. 

When  propositions  are  expressed  hypothetically,  then  con- 
verse propositions  are  such  that  the  hypothesis  of  each  is  the 
conclusion  of  the  other.  Every  good  definition  is  a  proposition 
whose  converse  is  true  ;  but  the  converse  of  a  true  statement 
may  be  false. 

Sometimes  the  hypothesis  of  a  theorem  is  complex,  i.  e.,  con- 
sists of  several  distinct  hypotheses  ;  in  this  case  every  theorem 
formed  by  interchanging  the  conclusion  and  one  of  the  hypo- 
theses is  a  converse  of  the  original  theorem. 

The  student  should  analyze  every  proposition,  separating  the 
subject  or  hypothesis  from  the  predicate  or  conclusion. 

A  Scholium  is  an  introductory  or  an  explanatory  remark. 

Axioms  that  apply  to  all  mathematics  are  called  general, 
to  distinguish  them  from  those  which  relate  to  geometrical  no- 
tions. The  statement  of  general  axioms  in  works  on  Geometry 
is  generally  defective,  and  it  is  unnecessary. 

Proportion. 

3,  The  following  properties  of  proportions  are  given  here 
without  demonstration.  The  student  is  referred  to  the  algebra 
for  a  fuller  treatment  of  this  subject. 

If  four  quantities  are  in  proportion  taken  in  their  direct  order,  then, 

I.  They  are  in  proportion  when  taken  alternately ; 

II.  They  are  in  proportion  when  taken  inversely ; 

III.  They  are  in  proportion  when  taken  by  composition ; 

IV.  They  are  in  proportion  when  taken  by  division ; 


Art.  3.]  PRELIMINARY.  3 

Y.  They  are  in  proportion  when  taken  by  composition  and  division. 

YI.  If  there  is  a  series  of  equal  ratios,  then  the  sum  of  the  antecedents 
is  to  the  sum  of  the  consequents  as  any  antecedent  is  to  its  consequent. 

YII.  The  products  of  the  corresponding  terms  of  two  proportions 
are  in  proportion. 

YIII.  The  quotients  of  the  corresponding  terms  of  two  proportions 
are  in  proportion. 

IX.  If  two  proportions  have  the  same  antecedents,  the  consequents 
are  in  proportion  in  their  direct  order. 

X.  If  two  proportions  have  the  same  extremes,  the  means  of  one 
proportion  may  be  substituted  for  the  extremes  of  the  other. 


CHAPTER  II. 
THE  SUBJECT  STATED. 

Article  1. — Every  material  object  occupies  a  portion  of 
space,  and  has  extent  and  form.  For  example,  this  book  occupies 
a  certain  space ;  it  has  a  definite  extent  and  an  exact  form.  These 
properties  may  be  considered  distinct  from  all  others.  If  the 
book  is  removed,  the  space  which  it  has  occupied  remains,  that 
is,  the  space  exists  independently,  and  this  space  has  extent  and 
form. 

Such  a  limited  portion  of  space  is  called  a  solid.  Be  careful 
to  distinguish  the  geometrical  solid,  which  is  a  portion  of  space, 
from  the  solid  body  which  occupies  space. 

The  limit  or  boundary  that  separates  a  solid  from  the  sur- 
rounding space  is  a  surface.  Every  surface  has  extent  and 
form,  but  a  surface  has  no  thickness  or  depth. 

Two  geometrical  solids  may  occupy  the  same  space,  entirely 
or  partially.  The  space  which  has  been  occupied  by  a  book 
may  be  occupied  by  a  block  of  wood.  Their  surfaces  may  meet 
or  cut  each  other. 

The  boundaries  of  a  surface  are  lines.  The  intersection  of 
two  surfaces,  being  the  limit  of  the  parts  into  which  each  divides 
the  other,  is  a  line.  A  line  has  extent  and  form,  but  neither 
breadth  nor  thickness. 

The  ends  of  a  line  are  points.  The  intersections  of  lines  are 
also  points.     A  point  has  neither  extent  nor  form. 

3.  A  Magnitude  is  that  which  has  only  extent  and  form. 
Geometry  is  the  science  of  magnitude. 
A  Solid  IS  a  magnitude  having  length,  breadth,  and  thickness. 
A  Surface  is  a  magnitude  having  length  and  breadth,  with- 
out thickness. 


Akt.  2.]  THE  SUBJECT  STATED.  6 

A  Line  is  a  magnitude  having  length,  without  breadth  or 
thickness. 

A  Point  has  only  position,  without  extent. 

A  line  may  be  measured  in  only  one  way,  that  is,  a  line  has 
only  one  dimension.  A  surface  has  two,  and  a  solid  has  three 
dimensions.  Every  magnitude  belongs  to  one  of  these  three 
classes. 

The  extent  of  a  line  is  called  its  length ;  of  a  surface,  its 
area ;  and  of  a  solid,  its  volume. 

The  word  magnitude  has  been  used  by  some  writers  in  the 
sense  of  quantity,  "  that  of  which  greater  or  less  can  be  predi- 
cated when  two  of  the  same  kind  are  compared  together,"  but 
generally  the  term  is  limited  to  lines,  surfaces,  and  solids. 

The  Postulates. 

3«  Postulate  of  Form. — A  magnitude  may  he  of  any 
form. 

Postulate  of  Extent. — A  magnitude  may  he  of  any  ex- 
tent. 

The  form  may  be  the  most  simple,  as  of  a  straight  line,  or  it 
may  be  as  complex  as  the  most  intricate  piece  of  machinery  that 
can  be  imagined. 

Lines,  surfaces,  and  solids,  may  be  contained  within  the 
limits  of  the  smallest  spot  that  represents  a  point,  or  they  may 
have  such  extent  as  to  reach  across  the  universe.  The  as- 
tronomer knows  that  his  lines  reach  to  the  stars,  and  that  his 
planes  extend  beyond  the  sun.  The  space  attributed  to  an  atom 
of  matter  is  divisible  into  any  number  of  smaller  spaces. 

The  postulates  do  not  assert  that  every  magnitude  can  be 
represented  to  the  senses  by  a  drawing  or  otherwise.  It  is  not 
asserted  that  every  magnitude  can  be  represented  even  in  im- 
agination ;  no  person  can  so  represent  the  magnitudes  of  as- 
tronomy or  of  physics,  although  convinced  of  their  reality.  The 
postulates  only  assert  that  whatever  magnitude  can  be  definitely 
apprehended  does  exist  ;  it  is  a  subject  of  geometrical  thought. 

The  form  of  a  magnitude  consists  in  the  relative  position  of 
its  parts.  However  the  extent  may  vary,  the  form  is  unchanged 
if  all  the  points  keep  the  same  relative  position. 


6  ELEMENTS  OF  GEOMETRY.  [Chap.  H. 

Similar  Magnitudes  are  those  that  have  the  same  form. 

Homologous  points  are  similarly  situated  points  in  similar 
magnitudes.  The  term  homologous  is  also  applied  to  similarly- 
situated  lines,  or  surfaces,  of  similar  magnitudes. 

Equivalent  Magnitudes  are  those  that  have  the  same 
extent. 

Superposition. 

4,  One  magnitude  may  be  mentally  applied  to  another  in 
order  to  compare  their  form  or  their  extent.  This  is  called  the 
method  of  superposition — one  magnitude  being  placed  upon 
the  other. 

Equal  Magnitudes  are  such  that  one  can  be  applied  to 
the  other,  and  the  two  coincide.  Since  they  coincide,  every 
part  of  one  has  its  corresponding  equal  part  in  the  other,  and 
the  parts  are  arranged  the  same  way  in  both.  Conversely,  if 
two  magnitudes  are  composed  of  parts  respectively  equal  and 
similarly  arranged,  one  may  be  applied  to  the  other,  part  by 
part,  till  the  magnitudes  coincide  throughout,  which  shows  them 
to  be  equal.  Hence,  equal  magnitudes  have  the  same  extent 
and  the  same  form ;  and  conversely,  if  magnitudes  are  both 
similar  and  equivalent,  they  are  equal. 

Two  magnitudes  composed  of  parts  respectively  equal,  but 
differently  arranged,  are  equivalent,  for  the  sums  of  equal  ex- 
tents are  equal ;  but  such  magnitudes  may  be  unequal,  that  is, 
they  may  not  coincide. 

Figures. 

5,  In  order  that  a  magnitude  may  be  a  subject  of  stated 
reasoning,  the  term  that  designates  it  must  be  definite,  its  mean- 
ing must  be  precise. 

A  Geometrical  Figure  is  any  line,  surface,  solid,  or 
combination  of  these  magnitudes  that  can  be  described  in  exact 
terms. 

Figures  are  represented  by  diagrams,  and  by  models.  In 
common  language,  such  representations  are  called  figures  ;  a 
small  spot  is  called  a  point,  and  a  long  mark  a  line,  but  they 
are  not  geometrical  points  and  lines.  They  have  not  only  extent 
and  form,  but  also  color,  weight,  and  other  properties. 


Art.   5.]  THE  SUBJECT  STATED.  7 

The  subject  of  a  proposition  is  said  to  be  given.  The  addi- 
tions to  the  figure  made  for  the  purpose  of  demonstration  or 
solution  constitute  the  construction. 

In  the  diagrams,  points  are  designated  by  capital  letters. 
Thus,  the  points  A  and  B  at  the  extremities  of  the  line. 

Figures  are  usually  designated  by  naming  some  of  their 
points,  as  the  line  AB,  and  the  quadrilateral  CDEF^  or  simply 
DF. 

A    C 


Sometimes,  when  a  figure  is  designated  by  a  single  letter,  the 
small  letters  are  used.     Thus,  the  line  m,  or  the  circle  n. 


Liines. 

6.  A  Straight  Line  is  one  that  has  the  same  direction 
through  its  whole  extent. 

The  path  of  a  point  moving  in  one  direction,  turning  neither 
up  nor  down,  right  nor  left,  is  a  straight  line. 

From  one  end  to  the  other,  a  straight  line  has  only  one  direc- 
tion ;  but  from  the  other  end  to  the 

first  it  has  exactly  the  opposite  di-         jy^ b 

rection.     A  straight  line  has  only 

two  directions.     The  direction  from 

^  to  J5  is  called  the  direction  AB^  and  the  direction  from  B  to 

A  is  called  the  direction  BA. 

A  Curve  is  a  line  that  has  a  continuous  change  in  direction. 

The  Curvature  of  a  line  is  the  amount  of  change  in  direc- 
tion. 

A  line  may  curve  so  as  to  return  to  the  same  point  within  a 
small  space,  or  the  change  may  be  so  slight  that  it  can  hardly  be 
distinguished  from  a  straight  line. 


8  ELEMENTS  OF  GEOMETRY,  [Chap.  II. 

Lines  may  have  a  uniform  curvature,  as  the  circmnference  of 
a  circle,  defined  in  Article  15  ;  or  the  curvature  may  vary,  as  in 
a  spiral,  an  ellipse,  or  a  reversed  curve. 


A  line  may  be  composed  of  parts  that  are  straight,  or  of 
curves,  or  of  both  curved  and  straight  parts.  A  line  composed 
of  straight  parts  is  called  a  broken  line. 


Straight  Liines.     Axioms. 

7.  Axiom  of  Direction, — In  one  direction  from  a  point, 
there  can  he  only  one  straight  line. 

Axiom  of  Distance. — The  straight  line  is  the  shortest  that 
can  join  two  points. 

Direction  is  a  term  that  is  too  simple  to  admit  of  definition. 
It  is  essentially  distinct  from  any  notion  of  extent.  The  direc- 
tion of  a  straight  line  is  determined  by  any  two  points  of  the 
line,  however  near  or  far  apart  they  may  be. 

8.  As  a  corollary  of  the  postulates,  we  have  this 
Problem. — There  may  he  a  straight  line  from  any  point,  in 

any  direction,  and  of  any  extent. 

Therefore,  there  may  be  a  straight  line  from  any  point  to 
any  other,  and  indefinitely  beyond  either. 

9.  From  the  Axiom  of  Direction  we  have  these 
Corollaries. — ^I.  From  one  point  to  another  there  can  be  only 

one  straight  line. 


Art.  9.]  THE  SUBJECT  STATED.  9 

II.  If  two  straight  lines  have  two  points  common,  or  if  they 
have  the  same  direction  from  a  common  point,  they  must  coin- 
cide ;  they  are  one  line. 

III.  The  position  of  a  straight  line  is  determined  by  two 
points,  or  by  one  point  and  one  direction. 

IV.  If  any  figure  revolves  about  a  straight  line,  the  line  it- 
self remains  fixed  in  position. 

This  property  is  peculiar  to  the  straight  line.  If  the  curve 
BG  revolves  upon  the  points  JB 

and  C,  the  straight  line  B  G  does  ^ ^--.^^^ 

not  move  while  the  curve  moves         B  ^- --^  c 

around  it. 

An  Axis  is  a  line  about  which  a  figure  revolves. 

Distance  implies  a  limited  extent,  the  shortest  path  from  one 
point  to  another. 

The  Axiom  of  Distance  states  that  property  of  a  straight 
line  which  was  used  by  Archimedes  as  the  definition.  He  has 
been  followed  by  Legendre,  one  of  the  most  distinguished  of 
modern  geometers.  The  defect  of  this  definition  is  that  it  makes 
straightness  a  matter  of  extent,  while  it  is  essentially  a  matter  of 
form. 

Euclid  defined  a  straight  line  as  "  one  that  lies  the  same  as 
to  the  points  in  it." 

A  fine  thread  being  stretched  till  it  assumes  that  position 
which  is  the  shortest  path  between  its  ends  is  a  good  representa- 
tion of  a  straight  line. 

Surfaces. 

10.  Surfaces  are  classified,  as  lines  are,  according  to  uni- 
formity or  change  of  direction.  At  one  point  of  a  line,  there 
are  only  two  directions  along  the  line,  but  at  one  point  of  a  sur- 
face there  are  an  indefinite  number  of  directions  along  the  sur- 
face. 

A  Plane  is  a  surface  that  never  varies  in  direction. 

A  Curved  Siirface  is  one  in  which  there  is  a  change  of 
direction  at  every  point. 

The  surface  of  a  small  body  of  still  water  is  perceptibly 
straight  in  every  direction.     A  piece  of  paper  may  be  bent  to 


10  ELEMENTS  OF  GEOMETRY.  [Chap.  II. 

illustrate  a  curved  surface  which,  from  any  point,  is  straight  in 
one  direction  and  curved  in  others.  A  globe  serves  to  illustrate 
a  surface  that  is  not  straight  in  any  direction. 

Planes. 

11.  Theorem. — If  two  planes  coincide  to  any  extent^  they 
must  coincide  throughout. 

This  follows  from  the  definition  of  a  plane.  If  two  planes 
could  have  a  common  portion  and  then  diverge,  some  direction 
of  one  of  them  would  vary. 

12.  Theorem. — A  straight  line  that  has  two  points  in  a 
plane  lies  wholly  in  the  plane. 

This  follows  from  the  definitions  of  straight  line  and  plane. 

Uniformity  of  direction  characterizes  both  the  plane  surface 
and  the  straight  line. 

Corollary. — That  part  of  a  plane  on  one  side  of  any  straight 
line  in  it  may  revolve  about  the  line  till  it  meets  the  other  part, 
when  the  two  coincide. 

It  is  to  be  understood  that  geometrical  lines  and  surfaces  are 
not  limited  in  extent,  unless  it  is  expressly  stated  or  implied  in 
the  context.  In  the  above  theorems,  if  either  of  the  magnitudes 
is  limited,  they  coincide  so  far  as  they  both  extend. 

Euclid  defined  a  plane  surface  as  "  one  that  lies  the  same  as 
to  the  straight  lines  in  it."  The  idea  is  that  of  invariable  direc- 
tion. Another  ancient  geometer  defined  a  plane  as  "  that  sur- 
face to  all  the  parts  whereof  a  straight  line  may  be  accom- 
modated."  This  is  in  substance  the  property  stated  in  the 
above  theorem.  It  is  the  definition  used  by  many  modern 
geometers. 

13.  Problem. — Through  any  three  points  there  may  be  a 
plane. 

Through  any  two  of  the  three  points  there  may  be  a  straight 
line  (8).  A  plane  may  pass  through  this  line,  and  if  the  third 
point  is  in  the  line  it  is  also  in  the  plane  (12).  If  the  third  point  is 
not  in  this  line,  the  plane  may  turn  on  the  line  as  an  axis  (9,  iv) 


Art.   13.]  THE  SUBJECT  STATED.  11 

until  it  passes  througli  the  third  point.     Then  all  three  points 
are  in  the  plane. 

Plane  Figures. 

14.  When  two  straight  lines  have  one  common  point  they 
lie  in  one  plane  (13),  and  from  this  point  the  lines  have  different 
directions  (7). 

An  Angle  is  the  difference  of  two  directions.  The  angle  is 
greater  or  less  according  as  this  difference  is  greater  or  less. 

Let  the  line  AB  be  fixed,  and  the  line  A  G   revolve  in  a 
plane  about  the  point  A,  taking  every 
direction  from  A  in  the  plane.     The  ^ 

angle  or  difference  in  direction  of  the  -^ 

two  lines  increases  2l8  AO  revolves. 

The  Arms  of  an  angle  are  the  lines 
that  form  it.  These  lines  have  been 
called  sides. 

Suppose  the  line  AC  revolves  about  the  point  A,  in  the 
same  plane  with  the  line  BD.     In  the 

course    of    one    revolution,   it    takes      ■ 

every  possible  direction  in  the  plane.                                       ^ 
One  of  these  is  the  direction  BD.  ' 

When  A  C  has  the  same  direction 
as  BI>,  then  the  direction  GA,  being  the  opposite  of  A  C,  is  the 
same  as  the  direction  DB,  which  is  the  opposite  of  BD. 

The  Vertex  of  an  angle  is  the  point  of  intersection  of  the 
arms. 

Parallel  Ijines  are  straight  lines  that  have  the  same  direc- 
tions. 

Angular  quantity  consists  in  the  difference  of  directions. 
Parallelism  consists  in  the  identity  of  directions. 

The  definition  of  similarity  can  now  be  stated  in  a  better 
form  than  the  one  given  in  Article  3. 

SimilsLT  Figures  are  such  that  every  possible  angle  in  one 
has  its  corresponding  equal  angle  in  the  other. 

15.  A  Plane  Figure  is  one  whose  points  all  lie  in  one 

plane  ;  for  example,  a  straight  line,  or  an  angle,  or  two  parallel 


13  ELEMENTS   OF  GEOMETRY.  [Chap.  H. 

lines.  None  of  the  plane  figures  just  mentioned  can  inclose  a 
portion  of  the  plane,  for  two  straight  lines  can  have  only  one 
common  point  (9). 

A  Polygon  is  a  portion  of  a  plane  bounded  by  straight 
lines.     The  straight  lines  are  the  sides  of  the  polygon. 

The  Perimeter  of  a  polygon  is  its  boundary,  or  the  sum  of 
all  the  sides. 

A  Triangle  is  a  polygon  of  three  sides.  A  quadrilateral 
has  four  sides  ;  a  pentagon,  five  ;  a  hexagon,  six  ;  an  octa- 
gon, eight ;  a  decagon,  ten ;  a  dodecagon,  twelve ;  and  a 
pentedecagon,  fifteen. 

The  line  AB  may  revolve  in  a 
plane  about  the  end  Aj  which  is  fixed. 
The  point  B  then  describes  a  curve 
which  returns  upon  itself.  Every 
point   of  this   curve   is   at  the  same 

distance  from  the  point  A,  this  dis-  \  ^b 

tance  being  equal  to   the   length   of 
AB, 

A  Circle  is  a  portion  of  a  plane  bounded  by  a  curve  that  is 
everywhere  equally  distant  from  a  point  within.  The  curve  is 
the  circumference,  and  the  point  is  the  center. 

Polygons  and  circles  are  by  their  definitions  plane  figures. 

Plane  Geometry  is  that  branch  of  the  science  which  treats 
of  plane  figures. 

In  this  work  plane  figures  are  discussed  in  the  following 
order  :  1.  The  straight  line  and  combinations  of  straight  lines 
(Chap.  Ill)  ;  2.  The  circumference,  and  straight  lines  in  com- 
bination with  it  (Chap.  IV)  ;  3.  Inclosed  rectilinear  figures,  that 
is,  the  triangle  (Chap.  Y),  the  quadrilateral  (Chap.  VI),  and 
other  polygons  (Chap.  VII)  ;  and  4.  The  circle  (Chap.  VIII). 

Figures  in  Space. 

16.  Geometry  of  Space  treats  of  figures  whose  points 
are  not  all  in  one  plane.  It  is  frequently  called  Solid  Geometry, 
but  strictly  a  solid  is  an  enclosed  figure. 

The  Figures  in  Space  which  are  treated  of  in  Elementary 
Geometry  are  :    1.  Unenclosed  figures  that  consist  of   straight 


Art.  16.]  THE  SUBJECT  STATED.  13 

lines  and  planes  (Chap.  IX)  ;  2.  Enclosed  figures  that  are 
bounded  by  planes,  and  are  called  polyedrons  (Chap.  X)  ;  and, 
3.  Three  inclosed  figures  bounded  by  curved  surfaces,  the  cone, 
the  cylinder,  and  the  sphere  (Chap.  XI). 

17.  Scholium. — This  chapter  contains  the  first  principles 
of  Geometry  and  the  definitions  of  the  most  important  terms  and 
figures. 

Four  fundamental  truths  are  the  basis  of  the  science — two 
postulates  and  two  axioms. 

Every  geometrical  conception,  however  simple  or  complex, 
is  composed  of  two  kinds  of  elements — directions,  and  lengths 
or  distances.  The  directions  determine  its  form,  and  the  dis- 
tances its  extent. 

18.  Exercises. — 1.  What  geometrical  principle  is  used  to  ascer- 
tain if  a  mark  is  straight,  when  one  applies  a  straight  edge  to  it  and 
observes  whether  they  coincide  ? 

2.  What  principle  is  used  to  ascertain  if  a  surface  is  plane,  when  one 
applies  a  straight  edge  to  it  in  many  positions,  observing  whether  it 
touches  throughout? 

3.  In  a  diagram  two  letters  suffice 
to  mark  a  straight  line,  but  it  may 
require  three  to  designate  a  curve. 
Why? 

4.  Which  is  the  greater  angle,  a  or 
5,  and  why? 

5.  What  is  the  greatest  number  of 
points  in  which  two  straight  lines  may 

cut  each  other  ?    In  which  tliree  may  cut  each  other  ?    Four  ? 

6.  What  principle  is  applied  when  a  stretched  cord  is  used  to  mark  a 
straight  line  ? 


PLANE  GEOMETRY. 


CHAPTER  III. 
STRAIGHT   LINES. 

Article  1. — Straight  lines  may  be  given  in  a  plane  in  any 
position,  and  witli  any  length  (II,  8),  and  either  separately  or 
combined. 

A  line  may  be  given  in  extent  only  :  for  example,  "  one 
inch,"  or,  "  a  line  equal  to  twice  AB^  A  line  may  be  given 
in  position  only  ;  as,  "  a  line  through  A  and  j5,"  or,  "  a  line 
parallel  to  AB  through  6V'  A  line  may  be  given  in  extent 
and  position  ;  as,  "  the  line  AB^''  the  ends  being  designated. 

I*TO'blem.. — Straight  lines  may  he  added,  subtracted,  multi- 
plied, or  divided. 

Straight  lines  may  be  of  any  length  (II,  8).  Therefore, 
there  may  be  a  line  equal  to  any  sum  or  to  any  difference  of 
other  lines,  and  there  may  be  a  line  equal  to  any  multiple  or  to 
any  equal  part  of  any  other  line,  or  having  any  ratio  to  it. 

A  line  may  be  thought  of  as  the  trace  of  a  point  in  motion. 
As  the  point  moves,  the  growing  line  has  every  length  from 
zero  to  the  entire  distance  passed  over.  It  has  an  infinite  num- 
ber of  various  lengths  which  no  drawing  could  exactly  represent, 
and  its  existence  at  every  one  of  those  lengths  is  as  certain  as 
that  of  a  geometrical  line  can  be. 

This  problem  does  not  assert  that  a  line  can  be  drawn  or 
otherwise  represented  equal  to  the  sum  of  two  other  lines,  or 
that  it  can,  by  dividers  or  other  instruments,  be  divided  into 
equal  parts,  etc.  This  problem  merely  asserts  the  possible  exist- 
ence of  such  sums,  parts,  etc.     Problems  in  drawing  may  require 


Art.  1.]  STRAIGHT  LINES.  15 

an  application  of  geometrical  principles,  but  such  work  is  not 
pure  geometry. 

Theorem. — Straight  lines  are  similar  figures. 

The  only  possible  difference  of  directions  in  a  straight  line 
is  the  same  as  in  any  other  straight  line.  For  each  of  them  has 
only  two  directions,  which  are  exactly  the  opposite  of  each 
other.  Straight  lines  have,  therefore,  the  same  form.  See  the 
definition  of  similar  figures  (II,  14). 

If  one  straight  line  is  applied  to  another,  they  coincide  so 
far  as  they  both  extend  ;  they  differ  only  in  extent. 

Corollary  I. — The  points  in  two  lines  which  divide  them  in 
the  same  ratio  are  ho- 
mologous points.  Thus,  j^  c  B 
if   the  lines   AB  and 

EI)  are  divided  at  the  ^ Z ^ 

points  C  and  F,  so  that 

AC '.AB  =  EF'.EB,  then  C  and  F  are  homologous  points  in 

these  similar  figures,  and  A  G  and  EF  are  homologous  parts. 

A  point  in  a  straight  line  is  said  to  divide  it  internally, 
or  simply  to  divide  it ;  and  a  point  in  the  line  produced  is  said 
to  divide  it  externally.  In  both  cases,  the  distances  of  the 
point  from  the  extremities  of  the  line  are  called  segments  of 
the  line. 

Thus,  the  line  AB  is  divided  internally  at  the  point  F,  and 
ext  era  ally  at  the  point 

S.     When   divided   at  P ^ s_^ 

S,  AS  and  BS  are  the       A  B 

segments. 

Corollary  II. — A  straight  line  is  equal  to  the  sum  or  to 
the  difference  of  its  segments,  according  as  it  is  divided  inter- 
nally or  externally. 

A  Secant  is  a  line  that  cuts  or  passes  through  any  other 
line  or  plane  figure. 

The  lengths  of  any  two  lines  of  definite  extent,  being  quan- 
tities of  the  same  kind,  have  a  certain  ratio. 

When  two  lines  have  a  common  measure,  their  ratio  can  be 
exactly  expressed  by  two  whole  numbers,  that  is,  by  the  num- 
bers that  tell  how  many  times  the  lines  contain  the  common 


16  PLANE  OEOMETR  Y.  [Chap.  III. 

measure.  Conversely,  if  the  lengths  can  be  expressed  by  two 
whole  numbers,  the  lines  are  commensurable,  that  is,  they 
have  a  common  measure.  Frequently  two  lengths  are  incom- 
mensurable, and  the  ratio  can  not  be  expressed  by  ordinary 
numerals.  Such  ratios  are  expressed  by  reference  to  the 
quantities  compared,  as  in  the  Corollary  I  above.  Radicals 
are  examples  of  numbers  incommensurable  with  unity. 

S,  A  line  is  said  to  be  divided  in  extreme  and  mean 
ratio  when  one  segment 

of  the  line  is  a  mean  pro-  ^  b  c 

portional     between     the 
whole  line  and  the  other 

segment.  Thus,  \i  AG  \  AB  =  AB  :  BC,  the  line  AG  h  dl 
vided  at  B  in  extreme  and  mean  ratio. 

The  student  of  algebra  may  prove  that  the  ratio  of  the 
greater  segment  to  the  less  is  ^(|/5  +  1).  It  will  be  shown 
in  the  sequel  how  to  mark  the  point  of  division  of  a  line  in  this 
ratio  (Y,  44,  x). 

Theorem.. — If  a  U?ie  is  divided  in  extreme  and  mean  ratio, 
the  segments  have  no  common  measure. 

Suppose  the  line  divided  at  B,  so  that  A  G :  AB  =  AB  :  BG. 
If  these  segments  have  a 

common  measure,  it  is  also  a        n         E     b  c    ^ 

a    common    measure     of 
their   difference  ;   that  is, 

if  BD  is  taken  equal  to  BG,  any  line  contained  an  in- 
tegral number  of  times  in  AB  and  in  BG,  or  its  equal  BI), 
must  be  contained  an  integral  number  of  times  in  AB,  for  the 
difference  of  two  whole  numbers  is  a  whole  number.  In 
the  same  way,  if  BJE  is  taken  equal  to  AB,  the  common 
measure  sought  must  be  a  measure  of  the  remainder  BJE,  And 
so  on,  the  common  measure  of  the  segments  AB  and  BC  must 
be  a  measure  of  every  remainder  as  long  as  there  is  a  remain- 
der found  by  such  subtractions. 

Since  BB  equals  B  G,  the  ratio  AB :  BD  is  the  same  as 
A  G :  AB.  Therefore,  there  was  necessarily  a  remainder  when 
BB  was  taken  from  AB,  and  AB  :  BB  =  BB :  AB,      That 


Art.  2.]  STRAIGHT  LINES.  17 

is,  AB  is  divided  in  extreme  and  mean  ratio.  For  the  same 
reason,  when  DE  (equal  to  AD)  is  taken  from  DB  there  must 
be  a  remainder,  and  the  ratio  DE\EB  is  the  same  as  the  first. 
The  part  taken  away  has  at  every  step  the  same  ratio  to  the  re- 
mainder. Therefore,  there  cannot  cease  to  be  a  remainder  ;  and, 
as  there  is  no  last  remainder,  the  segments  have  no  common 
measure. 

A  line  is  said  to  be   divided  harmonically  when  it  is 
divided  both  internally  and 
externally  in   the   same  ra- 

.  'PS 

tio.      For  example,   if  the  ■ ^ — ^ ' 

line  AB  is  divided  at  P  and 
at  /S,  so  that 

AP\AS=BP'.BB, 

then  AB  is  divided  harmonically. 

This  agrees  with  the  numerical  definition  of  harmonic  ratio, 
for  the  several  distances  from  A  to  P^  B^  and  S^  are  such  that 
"  the  first  is  to  the  third  as  the  difference  between  the  first  and 
the  second  is  to  the  difference  between  the  second  and  the 
third." 

Broken  Liiues. 

3.  A  curve  or  a  broken  line  is  said  to  be  concave  to  the 

side  on  which  there  can  be  a  straight  line  joining  two  of  its 
points,  and  convex  to  the  other  side.  Part  of  a  curve  or  of 
a  broken  line  may  be  concave  to  one  side  and  part  concave  to 
the  other  side. 

Theorem. — A  broken  linCy  no  part  of  which  is  concave 
toicard  another  line  that  unites  its  extreme  points,  is  shorter  than 
that  line. 

It  is  to  be  proved  that  ^ 

the  line  ABCD  is  shorter  ^/~y \^ 

than   the   line  AEGD,   to-  //b      ■ ____^       \ 

ward  which  no  part  of  the         /^  c  \Q^ 

first  line  is  concave.  D 

Produce  AB  and  ^(7 
till  they  meet  the  outer  line  in  F  and  JZ 
2 


18  PLANE  GEOMETRY.  [Chap.  m. 

Then  „ 

E, — i ,c 

AB+BF<AE+EF,  /X-~^^^  \ 

B  (7+  €n<  BF+Fa+  GH,      //^"^  ---  -  -■  -W 

and      CD<GH^-HD,  ^a  ^    \r 

T) 
Adding,  and  omitting  the 
common  terms, 

AB  +  BG^  CD  <  AE+FF+  FG  +  GH ^  HD. 

4.  Exercises. — 1.  Two  lines  being  divided  in  the  same  ratio  (see 
first  figure  in  this  chapter),  state  the  proportion  of  the  four  segments 
directly,  inversely,  and  by  alternation.  State  the  proportions  that 
result  by  composition,  and  by  division. 

2.  Show  that  the  sum  and  the  difference  of  two  lines  are  together 
twice  as  long  as  the  greater  line. 

3.  Show  that  if  AB  is  divided  harmonically  at  P  and  S^  then  PS  is 
divided  harmonically  at  A  and  B.     [I,  3,  i.] 

4.  If  the  line  -4^  is  divided  internally  in  extreme  and  mean  ratio  at 
the  point  P,  then  BC^  the  shorter  segment,  is  divided  externally  in  ex- 
treme and  mean  ratio  at  the  point  A. 

Angles. 

5.  Since  an  angle  is  the  difference  of  two  directions,  any  two 
lines  that  have  different  directions  from  a  common  point  form 
an  angle.  The  angles  considered  in  this  chapter  are  made  by- 
straight  lines  ;  in  the  following  chapter  (IV),  angular  quantity 
in  its  relation  to  curves  and  curvature  is  explained  ;  and  in  later 
chapters  (IX  and  XI),  the 

cases  of  angles  whose  arms 
do  not  lie  in  the  same  plane. 
Three  letters  may  be 
used  to  mark  an  angle,  the 
one  at  the  vertex  being  in 
the  middle,  as,  the  angle 
BA  G.     When  there  can  be 

no  doubt  what  angle  is  intended,  one  letter  may  answer,  as,  the 
angle  C, 


Art.  5,] 


STRAIGHT  LINES. 


19 


It  is  frequently  conve- 
nient to  mark  angles  with 
letters  placed  between  the 
arms,  as  the  angles  a  and 
b. 

Problem. —  There  may  he  an  angle  of  any  quantity. 

This  follows  from  the  definition  and  the  development  of  an- 
gular quantity  by  the  revolution  of  a  line  (II,  14).  As  the  re- 
volving line  may  have  any  direction  (II,  8),  the  angle  may  in- 
crease from  zero  to  any  quantity. 

Corollary. — An  angle  may  be  equal  to  any  sum  or  to  any 
difference  of  other  angles, 
and  an  angle  may  have  any 
ratio  to  another  angle  ;  that 
is,  angles  are  quantities  that 
may  be  compared,  added,  ' 
subtracted,  multiplied,  or 
divided.  

The  angle  ADC  i&  equal 
to  the  sum  of  the  other 
two,  and  ABB  is  the  difference  between  AB  C  and  BB  C. 


Angles  formed  at  One  Point. 

6.  The  difference  of  the  two  directions  AB  and  A  C  may 
be  estimated  in  either  of 
two  ways.  The  line  AG 
(II,  14)  may  revolve  in  the 
plane  on  either  side  from 
AB.  Thus  two  angles,  m 
and  n,  are  formed  by  the 
lines  AB  and  A  C. 

These  angles  are  said 
to  be  conjugate ;  each 
is  conjugate  to  the   other. 

When  it  is  not  specified  which  of  two  conjugate  angles  is  in- 
tended, the  less  is  always  to  be  understood. 

A  Reflex  angle  is  the  greater  of  two  conjugate  angles. 


20  PLANE  GEOMETRY.  [Chap.  III. 

A    Straight    angle    is         ^  ^  ^ 

one  whose  arms  have  exact-         ' ' ' 

ly    opposite    directions,    so 

that  they  form  one  straight  line,  as  AB  and  A  C. 

Adjacent  angles  are  two  angles  that  have  the  same  ver- 
tex and  one  common  arm  between  them.  Thus,  the  angles 
a  and  h  are  adjacent. 


One  line  meeting  another  at  some  other  point  than  its  ex- 
tremity, makes  with  it  two  angles,  which  are  adjacent,  as  the 
angles  i?Z>(7  and  j5i>i<: 

Vertical  angles  are  the  opposite  angles  formed  by  two  in- 
tersecting lines. 

A  Right  angle  is  formed 
when  one  line  meets  another, 
making  the  adjacent  angles 
equal. 

A  Perpendicular  to  a  . 

line  is  another  line  making 
a  right  angle  with  it. 

An  Oblique  line  is  one  that  is  neither  perpendicular  nor 
parallel  to  another  line  to  which  it  is  referred.  An  angle  that  is 
neither  a  right  angle  nor  a  multiple  of  a  right  angle  is  called 
oblique. 

An  Acute  angle  is  one  that  is  less  than  a  right  angle. 

An  Obtuse  angle  is  one  that  is  greater  than  a  right  angle, 
and  less  than  a  straight  angle. 

Complementary  angles  are  two  whose  sum  is  equal  to  a 
right  angle.     Each  is  the  complement  of  the  other. 

Supplementary  angles  are  two  whose  sum  is  equal  to  a 
straight  angle.     Each  is  the  supplement  of  the  other. 


Art.  '7.] 


STRAIGHT  LINES. 


21 


7 .   Theorem, — All  straight  angles  are  equal. 

For  any  straight  angle  may  be  placed  on  another,  and  the 
two  must  coincide,  for  the  arms  of  each  constitute  one  straight 
line  (II,  9,  ii). 

Corollaries. — I.  All  right  angles  are  equal,  for  a  right  angle 
is  half  of  a  straight  angle. 

II.  The  sum  of  all  the  successive  angles  formed  in  a  plane, 
on  one  side  of  a  straight  line,  is  equal  to  a  straight  angle,  or  to 


two  right  angles  ;  for  the  straight  line  may  coincide  with  the 
arms  of  a  straight  angle. 

III.  Conversely,  when  the  sum  of  several  successive  angles 
is  equal  to  two  right  angles,  the  extreme  arms  form  one  straight 
line.  For  the  sum  being  one  straight  angle,  the  extreme  arms 
may  coincide  with  the  arms  of  a  straight  angle  (II,  4). 

ly.  The  sum  of  all  the  suc- 
cessive angles  formed  in  a  plane 
around  a  point  is  equal  to  two 
straight  angles,  or  to  four  right 
angles.  For  instance,  the  sum  of 
two  conjugate  angles. 

V.  Complements  of  the  same 
or  of  equal  angles  are  equal. 

VI.  Supplements  of  the  same 
or  of  equal  angles  are  equal. 

VII.  The  supplement  of  an  obtuse  angle  is  acute,  and  con- 
versely. 

VIII.  The  greater  an  angle 
the  less  is  its  supplement. 

IX.  Vertical  angles  are  equal. 
For  both  AFC  and  DFB  are  sup- 
plements of  the  same  angle  AFD, 


22  PLANE  GEOMETRY.  [Chap.  III. 

In  geometrical  investigations,  the  right  angle  is  the  standard 
or  unit  of  angular  quantity.  The  right  angle  is  divided  into 
ninety  equal  parts,  called  degrees,  marked  thus  °.  The  six- 
tieth part  of  a  degree  is  a  minute,  marked  thus  ';  and  the 
Gixtieth  part  of  a  minute  is  a  second,  marked  thus  ". 

8.  Exercises. — 1.  Of  two  conjugate  angles,  one  is  equal  to  twice 
the  other ;  how  many  degrees  in  the  less  ? 

2.  Prove  that  the  hisectors  of  two  adjacent  supplementary  angles  are 
perpendicular  to  each  other. 


3.  Find  the  angle  between  the  bisectors  of  adjacent  complementary 
angles. 

4.  If  ^  is  any  angle,  prove  that  90°  +  A  and  90°  —  A  are  supple- 
mentary. 

5.  State  the  converse  of  Corollary  lY,  Article  7. 

6.  If  the  arms  of  an  angle  and  the  bisector  of  it  are  produced  beyond 
the  vertex,  the  bisector  also  bisects  the  vertical  angle.     [7,  ix.] 


Perpendicular  and  Oblique  Lines. 

.  9.  Theorem. — There  can  he  only  one  line  through  a  given 
point  perpendicular  to  a  given  straight  line. 

For,  since  all  right  angles  are  equal,  all  lines  lying  in  one 
plane  and  perpendicular  to  a  given  line  must  have  the  same 
direction.  Now,  through  a  given  point  in  one  direction  there 
can  be  only  one  straight  line. 

When  the  point  is  in  the  given  line,  this  theorem  must  be 
limited  to  one  plane. 


Art.  10.] 


STRAIGHT  LINES. 


23 


10.  Theorem, — If  a  perpendicular  and  several  oblique 
lines  extend  from  a  common  point  to  a  given  straight 
line : 

1.  The  perpendicular  is  shorter  than  any  oblique  line; 

2.  Two  oblique  lines  that  meet  the  given  line  at  equal  dis- 
tances from  the  perpendicular  are  equal/ 

3.  An  oblique  line  that  meets  the  given  line  at  a  greater  dis- 
tance than  another  from  the  perpendicular  is  longer  than  the 
other. 

AD  being  perpeDdicular  and 
AG  oblique  to  BE,  it  is  to  be 
proved  that  AD  is  shorter  than 
AC. 

Let  the  figure  revolve  upon 
BE  as  an  axis  (II,  12,  c),  until 
the  point  A  falls  upon  the  plane 
of  the  original  figure  at  the  point 
F,  so  that  AD  coincides  with  FD 
and  A  G  with  FG. 

Since   the   angle 


i/F 


Therefore, 
than  AGF. 


FDG    coin- 
cides with  ADG,  it  also  must  be  a  right   angle. 
ADF  is   a  straight  line   (7,  iii),  and  is  shorter 
AD,  the  half  of  ADF,  must  also 
be  shorter  than  AG,  the  half  of 
AGF. 

Secondly,  let  AD  be  the  per- 
pendicular, and  A  G  and  AE  the 
oblique  lines,  making  GD  equal 
to  DE.  

Let  that  portion  of  the  figure 
on  the  left  of  AD  turn  upon  AD. 
Since  the  angles  ADB  and  ADF 

are  equal,  DB  takes  the  direction  DF',  and,  since  i)  (7  and  DE 
are  equal,  the  point  G  falls  on  E.  Therefore,  A  G  and  AE  coin- 
cide (II,  9,  ii),  and  are  equal. 

Lastly,  if  DF  is  longer  than  DG,  it  is  to  be  proved  that 
AF  IS  longer  than  A  G, 


JJ 


^ERSITY 

OF 


24 


PLANE  GEOMETRY. 


[Chap.  III. 


^H 


On  the  line  DF  take  a  part 
DJE  equal  to  D  C,  and  join  AE. 
Then  let  the  figure  revolve  upon 
B  Gy  the  point  A  falling  upon  IT, 
and  the  lines  AD,  AE  and  AF 
upon  ^2>,  ^Je'  and  HF 

Now,  AEH  is  shorter  than 
^7^-£r  (3)  ;  therefore,  AE,  the 
half  of    AEH,  is   shorter  than 

^i^  the  half  of  AEH,  But  AC  \^  equal  to  ^^.  Hence, 
AF  is  longer  than  ^  (7,  or  ^j^,  or  any  line  from  A  meeting 
the  given  line  at  a  less  distance  from  D  than  DF. 

CorollSLries. — I.  Conversely,  two  equal  oblique  lines  extend- 
ing from  a  common  point  to  a  given  line  meet  it  at  equal  dis- 
tances from  the  perpendicular. 

II.  If  two  such  oblique  lines  are  unequal,  the  foot  of  the 
shorter  is  nearer  the  perpendicular. 

III.  The  distance  from  a  point  to  a  straight  line  is 
the  length  of  the  perpendicular,  for  distance  means  shortest 
path. 

IV.  Equal  oblique  lines  make  equal  angles  with  the 
perpendicular ;  for  CAD,  when  applied,  coincides  with 
DAE. 

V.  Equal  oblique  lines  make  equal  angles  with  the  given 
straight  line  ;  that  is,  A  CD  and  AED  are  equal. 

VI.  A  point  may  be  at  the  same  distance  from  two  points  of 
a  straight  line,  one  on  each  side  of  the  perpendicular  ;  but  it 
cannot  be  at  the  same  distance  from  more  than  two  points  of  one 
straight  line. 

VII.  Any  point  of  a  line  that  is  perpendicular  to  another  at 
its  midpoint  is  equally  distant  from  the  two  ends  of  the  second 
line. 


11,  Theorem. — If  a  line  is  perpendicular  to  another  at  its 
midpoint,  every  point  out  of  the  perpendicular  is  nearer  to 
that  end  of  the  line  that  is  on  the  same  side  of  the  perpen- 
dicular. 

If  BF\b  perpendicular  to  ^  C  at  its  midpoint  B,  then  it  is  to 


Art.  11.] 


STRAIGHT  LINES. 


25 


be  proved  that  D,  a  point  not  in 
BF,  is  nearer  to  C,  on  the  same 
side  of  the  perpendicular  as  D, 
than  it  is  to  A. 

Join  DA  and  D  C ;  also  join 
C  with  G,  the  point  where  1>A 
cuts  the  perpendicular.     Then, 


B 


But 


Da^GC=I)G-\-GA  =  DA, 
DC  <DG  +  Ga 


Therefore  DC  i^  less  than  DA. 

Corollary. — If  a  line  has  two  points  each  equidistant  from 
the  ends  of  another  line,  the  two  lines  are  perpendicular  to  each 
other,  and  the  second  line  is  bisected  (II,  9,  iii). 

The  two  points  may  be  on  the  same  side,  or  on  opposite  sides 
of  the  second  line,  or  one  may  be  on  it. 


12,  A  Locus  is  a  line,  or  lines,  or  a  surface,  or  surfaces,  every 
point  of  which,  and  no  other  point,  satisfies  an  assigned  condition. 
It  is  called  the  locus  of  the  points  that  satisfy  the  condition. 

Thus,  a  line  perpendicular  to  another  at  its  midpoint  is  the 
locus  of  those  points  in  the  plane  that  are  at  the  same  distance 
from  both  ends  of  the  bisected  line. 

In  Plane  Geometry,  the  loci  are  always  lines.  When  a  locus 
is  a  line,  it  may  be  regarded  as  the  path  of  a  point  moving  under 
the  assigned  condition.  Thus,  if  a  point  moves  in  a  plane,  all 
the  while  keeping  at  the  same  distance  from  two  given  points, 
then  its  path  is  perpendicular  to  and  bisects  the  line  joining 
those  points. 

The  Projection  of  one  line 
on  another  is  that  part  of  the 
second  which  lies  between  per- 
pendiculars on  it  from  the  ends 
of  the  first. 

Thus  DE  is  the  projection  of 
AB  on  CF. 

The  second  line  is  called  the  line  of  projection.    It  is 


c  D 


.:b^ 


26 


PLANE  GEOMETRY. 


[Chap.  HI. 


given  in  position,  but  is  indefinite  in  extent,  for  AB  could  not 
be  projected  on  CD  if  the  latter  terminated  at  D.  The  line 
projected  must  be  given  in  position  and  extent. 

When  one  end  of  the  pro- 
jected line  is  on  the  line  of  pro- 
jection, only  one  perpendicular 
is  needed.  Thus,  CD  is  the  pro- 
jection oi  AG  on  BE. 

The  projection  here  defined 
is  called  orthogonal,  when  it  is 
necessary  to  distinguish  it  from 
other  kinds  of  projections  used  in  Descriptive  Geometry. 


Bisected  Angle. 

13.  Theorem. — Every  point  of  a  line  that  bisects  an  angle 
is  equidistant  from  the  arms  of  the  angle. 

Let  B  CD  be  the  given  angle, 
and  A  C  the  bisecting  line.  The 
distances  of  the  two  arms  from 
any  point  A  of  that  line  are  meas- 
ured by  perpendiculars  to  the 
arms,  as  AF  and  AE. 

Let  that  part  of  the  figure  on 
the  left  of  CA  revolve  upon  CA 

as  an  axis.     Since  the  angles  B  CA  and  D  CA  are  equal,  the  line 
CB  falls  upon  CD. 

Then  the  perpendiculars  AF  and  AE  coincide  (9),  and  the 
point  F  falls  upon  E.  Therefore,  AF  and  AE  are  equal,  and 
the  point  A  is  equally  distant  (10,  iii)  from  the  arms  of  the 
angle. 


14,  Theorem.. — Every  point  not  in  the  line  bisecting  an 
angle  is  nearer  to  that  arm  of  the  angle  that  is  on  the  same  side 
of  the  bisector. 

It  is  to  be  proved  that  the  point  G  is  nearer  to  the  arm  CD, 
which  is  on  the  same  side  of  the  bisector  as  G,  than  it  is  to  the 
arm  J5a 


Art.  14.] 


STRAIGHT  LINES, 


27 


Make  6^^  and  6^i^  per- 
pendicular respectively  to 
the  arms  CI>  and  J5C. 
From  the  point  A,  where 
GF  cuts  the  bisector, 
make  AH  perpendicular 
to  CD,  and  join  GH. 
Then 

GA  +  AH 

But 


GF, 
GE<  GH<  GA  +  AH 


Therefore  GF  is  less  than  GF 

It  may  be  that  the  bisector  is  not  cut  by  either  of  the  per- 
pendiculars from  the  point  to  the  arms  of  the  angle.  The  de- 
monstration of  that  case  is  left  to  the  student. 

Corollaries. — I.  If  a  line  passes  through  two  points,  each  of 
which  is  equally  distant  from  the  arms  of  an  angle,  the  line  bi- 
sects the  angle  (II,  9,  iii). 

11.  A  line  that  bisects 
an  angle  is  the  locus  of 
the  points  that  are  equally 
distant  from  the  arms. 
The  locus  of   the  points  jf 

equally  distant  from  two 
lines  that  cut  each  other 
is  the  two  lines  that  bi- 
sect the  angles  made. 
Thus,  FG  and  HF  are 
the  locus  of  the  points  equally  distant  from  A  C  and  DF, 

15.  Application. — Perpendicular   lines   are   constantly  used   in 
architecture,   carpentry,  ma- 
chinery, etc. 

The  instrument  called  a 
square  consists  of  two  pieces 
of  wood,  iron,  or  steel,  having 
straight  edges;  the  edges  of 
one  piece  being  at  right 
angles  to  those  of  the  other, 
lines  or  surfaces  are  required. 


It  is  used  in  any  work  where  perpendicular 


28  PLANE  GEOMETRY.  [Chap.  III. 

In  order  to  test  the  square,  draw  an  angle  with  it  on  any  plane  sur- 
face, as  BA  G.  Extend  BA  in  tlie  same 
straight  line  to  D.  Then  turn  the  square 
so  that  the  edge's  by  which  BAG  was 
drawn  may  be  applied  to  the  arms  of 
the  angle  DAG.  If  the  coincidence  is 
exact,  the  square  is  correct  as  to  these        j^ 


16.  Exercises. — 1.  Can  two  lines  have  the  same  projection  on  a 
third  ?  Wliat  is  the  projection  of  a  line  that  is  perpendicular  to  the  line 
of  projection  ? 

2.  What  is  the  geometrical  principle  involved  in  the  method  of  testing 
a  square  described  in  the  preceding  Article  ? 


Parallels. 

17.  Parallel  lines  are  straight  lines  that  have  the  same 
directions. 

CoroUeLiy. — Lines  that  are  parallel  to  the  same  line  are  par- 
allel to  each  other. 

18.  Theorem. — Through  a  given  point  there  can  he  only  one 
line  parallel  to  a  given  line. 

For,  in  one  direction  from  a  point  there  can  be  only  one 
straight  line. 

Corollary. — Parallel  lines  can  never  meet. 

19.  Theorem. — Two  parallel  lines  lie  in  one  plane. 
Let  AG   and  B-D   be   parallel 

lines.     Now,  there  may  be  a  plane  ^  ^ 

through  the  points  A,  IB,  and  C  (II,  ' 


13).      The  line  AG  lies  wholly  in  ^^  jy 

this  plane  (II,  12).     There  may  be  ~ 

in    this    plane    a    line    through   3 

parallel  to  A  G  (II,  14)  ;  but  there  can  be  through  J3  only 
one  line  parallel  to  A  G.  Therefore,  BD  is  that  line  in  the 
plane  of  A,  B,  and  G. 

Parallel  lines  were  defined  by  Euclid  as  "  straight  lines  which. 


Art.  19.]  STRAIGHT  LINES.  29 

being  in  the  same  plane,  and  being  produced  indefinitely  to  both 
sides,  meet  on  neither  side."  It  has  just  been  demonstrated  that 
these  properties  belong  to  two  straight  lines  that  have  the  same 
directions.  The  objections  to  the  definition  of  Euclid  are  that 
it  is  double,  and  one  of  the  attributes  is  negative.  The  defini- 
tion based  on  direction  was  introduced  by  an  American  geome- 
ter, James  Hayward. 

30,  When  two  straight  lines  in  the  same  plane  are  cut  by 
a  third,  the  angles  are  named  as  follows  : 

Corresponding  angles  are  any  two  having  their  vertices 
at  different  points,  both  being  on  the  same  side  of  the  secant, 
and  on  the  same  side  of  the  two  lines  cut ;  for  example,  h 
and  h. 

Alternate   angles   are   any  two   having  their  vertices   at 
different  points  and  being  on  op- 
posite  sides    of    the   secant   and 
on  opposite  sides  of  the  two  lines 
cut ;  for  example,  f  and  7c. 

Interior  angles  are  those 
between  the  two  lines,  as  /,  g,  A,  ^-^^         ^"\^       \ 

and  k.     The   others   are    exte- 
rior. 

Corollary. — The  corresponding  and  the  alternate  of  any 
given  angle  are  vertical  to  each  other,  and  therefore  equal.  For 
instance,  h  is  the  corresponding  of  A,  and  g  is  the  alternate  of  h. 

21.   Theorem. —  When  the  arms  of  an  angle  are  parallel  to 
those   of  another^  and  have  re- 
spectively   the    same    directions 
from  their  vertices,  the  angles  are 
equal. 

If  the  directions  iTi^ and  LH 
are  the  same  and  the  directions 
-ffZ>  and  LB  are  the  same,  the 
angles  FKD  and  HLB  are  equal, 
for  each  is  the  difference  of  the 
same  directions. 


30 


PLANE  GEOMETRY, 


[Chap.  III. 


33.  Theorem. —  Whoi  two  parallel  lines  are  cut  hy  a  secant^ 
each  of  the  eight  angles  is  equal  to  its  corresponding  angle. 

If  the  parallels  AB  and  CD 
are  cut  by  the  secant  EF^  the 
same  reasoning  as  in  the  last 
demonstration  proves  that  the 
angles  FKJD  and  KIB  are  equal. 
The  same  of  any  two  correspond- 
ing angles. 

Corollaries. — I.  If  two  angles 
have  their  arms  respectively  par- 
allel  and  the   directions  of   one 

angle  opposite  to  those  of  the  other,  the  angles  are  equal.  Thus 
the  angles  OKI  and  BLN  are  equal,  for  one  is  vertically  oppo- 
site to  an  angle  that  is  equal  to  the  other. 

II.  When  two  parallel  lines  are  cut  by  a  secant,  each  of  the 
eight  angles  is  equal  to  its  alternate. 

III.  If  two  angles  have  their  arms  respectively  parallel,  and 
one  pair  of  parallel  arms  have  opposite  directions  from  their 
vertices,  and  the  other  pair  have  the  same  directions  from  their 
vertices,  the  angles  are  supplementary.  Thus  the  angles  CKF 
and  AIj  G  are  supplementary. 

lY.  Two  interior  angles  on  the  same  side  of  a  secant  are 
supplements  of  each  other. 

V.  When  a  line  is  perpendicular  to  one  of  two  parallels,  it 
is  perpendicular  to  the  other,  and  all  the  angles  are  right. 


23.  Theorem.— When  two 
lines  in  07ie  plane  are  not  parallel, 
and  are  cut  hy  a  secant,  the  cor- 
responding angles  are  unequal. 

If  KN  is  not  parallel  to  AB, 
and  if  they  are  cut  by  FF  at  G 
and  11^  then  through  G  there  may 
be  a  line  CB  parallel  to  AB  (II, 
8). 

Since  CD  is  parallel  to  AB 
and  KJSF  is  not,  the  lines  CD 


AST.  23.] 


STRAIGHT  LINES. 


31 


and  jOT  make  an  angle  with  each  other  at  G.  Therefore, 
the  angles  which  they  make  with  UF  are  not  equal.  For 
instance,  EGN  is  less  than  EGD ;  but  EGD  is  equal  to 
its  corresponding  angle  EHB,  Therefore  EGN  is  less  than 
EHB,  and  so  it  may  be  proved  that  any  corresponding  angles 
made  by  AB  and  KJ^  with  EF  are  unequal. 


24,  Theorem. —  When  two 
straight  lines  in  the  same  plane 
are  cut  l)y  a  third,  making  the 
corresponding  angles  equal,  the 
two  lines  so  cut  are  parallel. 

For,  ii  GO  and  HA  were 
not  parallel,  the  corresponding 
angles  would  be  unequal  (23). 

Corollaries. — I.  If  the  alter- 
nate angles  are  equal,  the  lines  are  parallel. 

II.  The  lines  are  parallel  when  the  interior  angles  on  the 
same  side  of  the  secant  are  supplementary. 


Angles  with  Perpendicular  Arms. 

25.  Theorem. — Two  angles  that  have  their  arms  respective- 
ly perpendicular  are  equal  or  supplementary. 

If  AB  is  perpendicular  to  I)  G,  and  BC  is  perpendicular 
to  EF,  then  it  is  to  be  proved 
that  the  angle  ABC  is  equal 
to  one  and  supplementary  to  the 
other  of  the  angles  formed  by 
BG  and  EF. 

Make  BI  parallel  to   GB,  -^x 

and  BIT  parallel  to  EF. 

The  right  angles  ABI  and 
CBH  are  equal  (22,  v).  Sub- 
tracting the  angle  HBA  from 
both,  the  remainders  HBI  and 
AB  C  are  equal.  But  HBI  is 
equal  to  FGB  (21),  which  is  the  supplement  of  EGD,     There- 


33 


PLANE  GEOMETRY, 


[Chap.  III. 


fore,  the  angle  ABC  is  equal  to 
one  and  is  supplementary  to  the 
other  of  the  angles  formed  by  the 
lines  Z>  6^  and  ^i^  ^, 

Corollary. — When  the  vertex 
of  each  angle  is  between  the  arms 
of  the  other,  or  between  the  exten- 
sions of  those  arms  produced  be- 
yond the  vertex,  the  angles  are 
supplementary.  Also,  if  the  an- 
gles have  their  vertices  at  the  same  point,  and  both  the  arms 
of  one  are  between  the  arms  of  the  other,  the  angles  are  sup- 
plementary.    When  otherwise  placed,  the  angles  are  equal. 


B 


M 


D 


Distances  between  Parallels. 

26.  Theorem.. — Two  parallel  lines  are  everywhere  equally 
distant. 

A  line  perpendicular  to  one  of 
the  parallels  at  any  point  of  it  is 
also  perpendicular  to  the  other,  as 
L3I.     That  part  of  the  parallels  on  — 

one  side  of  LM  may  be  made  to 
revolve  on  L3I  as  an  axis.      As 

the  angles  at  L  are  equal,  LA  falls  upon  LB,  and,  as  the  an- 
gles at  M  are  equal,  31 G  falls  upon  3IB. 

Since  the  axis  L3f  may  be  placed  at  any  point,  it  follows 
that  any  part  of  the  parallels  may  be  made  to  coincide  with  any 
other  part.  Therefore,  the  distance  between  the  lines  is  every- 
where the  same. 

CoroUebry. — The  parts  of  parallel  lines  included  between 
perpendiculars  to  them  are  equal.  For  the  perpendiculars  are 
parallel. 

Secants  and  Parallels. 

27-  Theorem. —  When  several  parallel  lines  are  cut  hy  a 
secant,  if  one  distance  between  two  parallels  is  equal  to  another 
distance  between  two  parallels,  then  the  segments  of  the  secant  are 
equal. 


Art.  21.] 


STRAIGHT  LINES. 


33 


n            V 

G 

\o 

K 

\rs           ^ 

c 


H 


If  the  distance  between  the  parallels  B  and  D  is  the  same  as 
between  the  parallels  D  and  G,  it  is  to  be  proved  that  the  seg- 
ments JEI  and  10  are  equal. 

That  part  of  the  figure  be-  ^ 

tween  BC  and  BF  may  be 
placed  upon  and  coincide  with 
the  part  between  BF  and  GH. 
Since  two  parallels  are  every- 
where equally  distant,  they  may 
be  so  placed  that  the  point  F 
falls  upon  I.     Then,  since  the  \ 

angles  BEI  and  BIO  are  equal  -^ 

(22),  the  line  ^Z  takes  the  direc- 
tion 10,  and  since  BF  falls  on  GB^,  the  point  I  falls  on  0. 
Therefore,  EI  and  10  coincide  and  are  equal.     In  like  manner 
it  is  proved  that  any  two  of  the  segments  of  the  line  A  Y  are 
equal,  if  the  distances  between  the  parallels  are  equal. 

Corollaries. — I.  Conversely,  when  several  parallels  are  cut 
by  a  secant,  if  the  segments  of  the  secant  are  equal,  the  distances 
between  consecutive  parallels  are  equal. 

II.  When  several  parallels  intercept  equal  parts  of  a  secant, 
then  any  other  secant  is  divided  into  equal  segments  by  the 
same  parallels. 


38,  Theorem. —  When  three  parallel  lines  are  cut  by  two 
secants,  the  secants  are  divided  proportionally. 

If  AB,  BE,  and  CF  are  parallel,  it  is  to  be  proved  that 
AB:BC=zBE'.EF. 

When  AB  and  B  C  have  a 
ratio  that  can  be  expressed  by 
two  whole  numbers,  they  may 
be  divided  into  equal  parts 
corresponding  in  number  to 
the  terms  of  the  ratio.  Thus, 
m  and  n  being  integers,  if 
AB  :BC  =  m'.n,  AB  may  be 
divided  into  m  equal  parts,  and 
BG  into  n,  and  all  the  m  -{■  n  parts  are   equal.     Then  lines 


34  PLANE  GEOMETRY.  [Chap.  III. 

througli  the  points  of  division  parallel  to  AD  divide  DF  into 
equal  parts  (27,  ii).  The  number  of  parts  in  DJE  is  the  same  as 
in  AB^  and  the  number  in  EF  is  the  same  as  in  B  G.  Hence, 
AB:BC  =  DF:FF. 

When  the  ratio  AB  :  B  C  can  not  be  expressed  by  whole 
numbers,  these  segments  have  no  common  measure.     Any  meas- 
ure of  AB,  applied  to  B  (7,  leaves  a  part  unmeasured,  which  is 
less  than  the  measure.     Say  the  part 
measured  is  BG.     A  second  measure 
of  AB  may  be  taken  less  than  the 
remainder  G  C.    Applying  this  to  B  G, 
the  remainder  is  less  than  the  measure, 
and  therefore  less  than  G  G.     Using  a 
measure  regularly  smaller  than  the  re- 
mainder, the  part  oi  BG  measured  by 
a  measure  of  AB  is  a  constantly  in- 
creasing line  and  may  become  as  nearly 

equal  to  BG  as  we  choose.  For  a  measure  of  AB  may  be 
taken  smaller  than  any  line  that  can  be  assigned  (Postulate  of 
Extent). 

Let  BJT  represent  this  increasing  line,  let  a  line  parallel 
to  AB  pass  through  JT,  and  let  Y'  be  the  point  where  this  par- 
allel is  cut  by  BF.  As  shown  above,  since  AB  and  B^ 
have  a  common  measure,  AB  :  BJl  =  BE\  FY.  These  ratios 
are  variable,  but  always  equal.  As  the  lines  BX  and  FY  can 
approximate  the  limits  BG  and  FF,  the  ratio  AB :  BJl 
can  approximate  AB :  B  G,  and  the  ratio  BF :  FY  can  ap- 
proximate BF :  FF,  within  less  than  any  assignable  differ- 
ence. 

Now  the  constant  ratios  AB  :  B  G  and  BF :  FF,  which 
limit  the  variable  ratios,  must  be  equal.  For  if  there  were  any 
difference,  then  one  of  these  two  variables  could  exceed  the 
lesser  constant  ratio,  while  the  other  variable  could  not.  That 
is,  equals  could  be  unequal,  which  is  absurd.     Therefore 

AB:BG  =  BF',FF. 

Corollary.— By  alternation,  AB  :  BF  =BG:  FF.  Hence, 
when  any  number  of  parallels  are  cut  by  two  secants,  every 


Art.  28.]  STRAIGHT  LINES.  35 

segment  of  one  secant  has  tlie  same  ratio  to  the  corresponding 
segment  of  the  other. 

A  Variable  is  a  magnitude,  or 
other  quantity,  that  varies  according 
to  any  rule  made  for  the  purpose  of 
the  investigation. 

A  Constant  is  a  magnitude,  or 
other  quantity,  that  remains  un- 
changed. 

The  Ijimit  of  a  variable  is  a  con- 
stant that  the  variable  may  approxi- 
mate indefinitely,  but  can  never  quite  reach. 

Thus  the  line  ^  C  is  the  limit  of  BX,  The  law  govern- 
ing the  variation  of  BX  is  that  it  is  a  part  of  J3  G,  commensur- 
able with  AB^  and  that  the  common  measure  used  at  every  step 
is  less  than  the  last  remainder,  BC — BJC.  A  variable  may 
have  zero  for  its  limit.  For  example,  the  remainder  XC  may 
become  less  than  any  assignable  distance. 

Lines  not  Parallel  meet. 

39.  Theorem.— If  two  straight  lines  that  lie  in  one  plane 
are  not  parallel,  they  meet  when  produced. 

If  AB  and  CD  lying  in 
the  same  plane  are  not  par- 
allel, it  is  to  be  proved  that  "^^^-^ 

when   produced  they  must  ^  — -^^TTr::;^^. _^ 

™eet.  ^ Vi^^L^. „...^ 

Through  H  and  Z,  two  K  \      l"-^^^^^^    ^ 

points   of  AB,  make  UF  \  ^ 

and   JSTL    parallel  to   CD.  — ^ 

Make  IIG,  joining  the  two 
outer  parallels,  and  there- 
fore cutting  the  inner  parallel. 

Then,  as  just  demonstrated,  UK  is  to  KG  as  HI  is  to  the 
segment  of  AB  that  lies  between  the  parallels  NL  and  CD. 
Therefore,  at  a  definite  distance  from  ij  the  lines  AB  and  CD 
must  meet  and  cut  each  other. 


36  PLANE  GEOMETRY.  [Chap.  III. 

Corollary. — ^If  two  straight  lines  are  in  the  same  plane  and 
can  never  meet,  they  are  parallel. 

Scholium. — The  following  four  propositions  about  two 
straight  lines  that  lie  in  one  plane  have  a  relation  to  each  other 
that  deserves  attention  : 

1.  If  the  two  lines  are  parallel,  they  never  meet ; 

2.  If  the  lines  can  not  meet,  they  are  parallel ; 

3.  If  the  lines  can  meet,  they  are  not  parallel ;  and 

4.  If  the  lines  are  not  parallel,  they  can  meet. 

The  first  is  a  corollary  from  the  definition  of  parallels 
and  the  Axiom  of  Direction.  The  second  is  the  converse 
of  the  first,  but  is  not  a  consequence  of  it  and  requires  a 
separate  demonstration.  The  third  is  a  consequence  of  the 
first,  and  these  two  are  called  contrapositives.  Each  is  the 
contrapositive  of  the  other.  The  second  and  fourth  are  con- 
trapositives. 

Contrapositive  propositions  are  such  that  the  hy- 
pothesis of  each  consists  in  the  denial  of  the  conclusion 
of  the  other.  Either  of  two  contrapositives  is  a  corollary 
of  the  other.  Articles  23  and  24  ,of  this  chapter  are  con- 
trapositives. 


30.  Applications. — The  instrument  called  the  T-square  consists 
of  two  straight  rulers  at  right  angles  to  each 

other,  as  in  the  figure.     It  is  used  to  draw  ^         , 

parallel  lines.  Lay  the  cross-piece  of  the  in- 
strument along  a  straight  line  perpendicular  to 
the  direction  of  the  intended  parallels.  The 
other  piece,  called  the  blade,  may  be  moved, 
keeping  the  cross-piece  coincident  with  the 
perpendicular,  and  lines  parallel  to  each  other 
may  be  drawn  along  the  blade. 

The  draughting  instrument  called  a  trian- 


gle is  a  flat,  triangular  piece  of  wood  or  other  material,  right  angled  at 
one  corner.  It  is  used  for  drawing  parallel  lines  in  the  same  way  as 
the  T-square ;  but,  instead  of  a  straight  line  for  direction,  it  is  better  to 
use  a  ruler  held  firmly  to  the  paper. 

The  uniform  distance  of  two  parallels  is  a  common  principle  in  manu- 
factures and  the  mechanic  arts.    It  is  constantly  employed  in  the  con- 


Art.  30.]  STRAIGHT  LINES.  37 

struction  of  houses,  furniture,  and  macbinerj.  One  of  the  simplest  tools 
made  on  this  principle  is  the  joiner's  gauge,  used  to  draw  a  line  on  a 
board  parallel  to  its  edge. 

31.  Scholium. — Demonstration  by  the  method  of  superposi- 
tion has  been  used  several  times  in  this  chapter.  There  is  a  dif- 
ferent motion  required  in  different  cases,  to  place  one  figure  on 
another. 

Every  plane  figure  may  be  regarded  from  either  side.  Thus 
it  has  two  faces.  The  upward  face  or  obverse,  and  the  down- 
ward face  or  reverse. 

There  are  two  methods  of  superposition  of  plane  figures  :  the 
first,  called  direct,  when  the  reverse  of  one  figure  is  applied  to 
the  obverse  of  the  other,  and  the  second,  called  inverse,  when 
the  obverse  faces  are  applied  to  each  other.  In  the  former  case, 
of  which  there  is  an  example  in  Article  27,  one  figure  may  be 
supposed  to  slide  along  the  plane  till  the  coincidence  takes  place. 
In  the  latter  case,  of  which  there  are  examples  in  Article  10,  one 
figure  or  part  of  a  figure  must  be  turned  upon  an  axis  in  order 
that  its  obverse  may  be  applied  to  the  obverse  of  the  other  figure 
or  part  of  the  figure. 

32.  Miscellaneous  Exercises. — 1.  What  is  the  principle  in- 
volved in  the  use  of  the  T-square  ? 

2.  Find  the  locus  of  those  points  that  are  at  a  given  distance  from  a 
straight  line  given  in  position.  [Represent  the  given  line  by  a  drawing ; 
mark  many  points,  all  at  the  same  given  distance  from  the  line ;  then  de- 
scribe the  location  of  all  such  points.] 

3.  If  a  straight  line  joining  two  parallel  lines  is  bisected,  another  line 
through  the  point  of  bisection,  and  joining  the  two  parallels,  is  also  bi- 
sected at  that  point.  [Make  a  third  parallel  through  the  point  of  bisec- 
tion ;  apply  Corollary  II  of  Article  27.] 

4.  What  is  the  greatest  number  of  points  in  which  seven  straight  lines 
can  cut  each  other,  three  of  them  being  parallel  ? 

5.  If  two  lines  have  a  common  measure,  they  have  a  common  multi- 
ple ;  and  conversely. 


CHAPTER  ly. 

CIR  C  UMFER  ENCES, 

Article  1. — It  follows,  from  the  definition  (II,  15),  that  a 
circumference  of  a  circle  is  the  locus  of  all  the  points  of  a  plane 
that  are  at  the  same  distance  from  one  point, 

A  Radius  is  a  straight  line  from  the  center  to  the  circum- 
ference. 

A  Diameter  is  a  straight  line  through  the  center  with  both 
ends  in  the  circumference. 

Corollaries. — I.  Radii  of  the  same  circle  are  equal. 

II.  A  diameter  is  equal  to  two  radii. 

III.  Every  point  of  the  plane  is  outside  of,  on,  or  inside  of 
the  circumference,  according  as  its  distance  from  the  center  is 
greater  than,  equal  to,  or  less  than  the  length  of  the  radius. 

IV.  Circles  having  equal  radii  are  equal. 
Concentric  circles  are  those  having  the  same  center. 

2.  Theorem. — Three  points  not  in  the  same  straight  line  de- 
termine the  position  and  extent  of  a  circumference. 

To  demonstrate  this,  it  must  be  shown  that  one  circumfer- 
ence,  and    only    one,   may   be 

made  through  three  such  points,  jg 

A,  B,  and   G.     Join  AB  and  jrj ,.-"""    \ 

BC.     At  D  and  E,  the  mid-        A„^-"""\  \,f 

points  of  those  lines,  let  per-  A  y'    \   ^ 

pendiculars  be   erected  m   the  \    /  -^ 

plane  passing  through  the  three  /'\p 

points. 

Since  AB  and  B  0  have  dif- 
ferent  directions,  BG  is  not   perpendicular  to   BC.     Hence 


Art.  2.]  CIRCUMFERENCES.  39 

D  G  and  JEH  are  not  parallel,  and  they  must  meet  if  produced 
(III,  29) 

Since  every  point  oi  DG  is  equidistant  from  A  and  S  (III, 
10,  Yii),  and  every  point  of  JS H  is  equidistant  from  £  and  (7, 
their  common  point  L  is  equidistant  from  A,  B,  and  G.  There- 
fore, with  this  point  as  a  center,  a  circumference  may  be  de- 
scribed through  A^  B^  and  G,  There  can  be  no  other  circum- 
ference through  these  three  points,  for  there  is  no  other  point 
besides  L  equally  distant  from  all  three  (III,  11). 

Corollary, — Two  circumferences  can  cut  each  other  in  two 
points  only. 

3.  Theorem. — A  circumference  is  curved  throughout. 

For  a  straight  line  can  not  have  more  than  two  points  equally 
distant  from  a  given  point  (III,  10,  vi). 

Corollary. — A  straight  line  may  cut  a  circumference  in  two 
points  only. 

An  Arc  is  a  portion  of  a  curve. 

A  Chord  is  a  straight  line  joining 
the  ends  of  an  arc. 

4.  Theorem. — A  diameter  is  lon- 
ger than  any  other  chord  of  the  same 
circumference. 

For  any  other  chord  is  shorter  than 
the  sum  of  two  radii  (Axiom  of  Dis- 
tance). 

Symmetry  of  the  Circumference. 

5.  A  Center  of  Symmetry  is  a  point  with  reference  to 
which  other  points  are  symmetrical.  Two  points  are  symmet- 
rical with  reference  to  a  center  of  symmetry  when  it  is  the  mid- 
point of  the  straight  line  that  joins  them.  For  example,  the 
ends  of  a  diameter  are  symmetrical  with  reference  to  the  center 
of  the  circle. 

An  Axis  of  Symmetry  is  a  straight  line  with  reference  to 
which  points  are  symmetrical.     Two  points  are  symmetrical  with 


40  PLANE  GEOMETRY.  [Chap.  IV. 

reference  to  an  axis  of  symmetry,  when  the  line  that  joins  them 
is  bisected  perpendicularly  by  the  axis. 

There  is  also  a  plane  of  symmetry,  but  its  consideration  is 
deferred. 

Two  figures  are  symmetrical  with  reference  to  a  center  or 
to  an  axis  of  symmetry,  when  every  point  in  each  has  its  symmet- 
rical point  in  the  other.  A  symmetrical  figure  is  one  that  can  be 
divided  into  two  parts  that  are  symmetrical  with  reference  to 
an  axis  of  symmetry.  Any  straight  line  is  a  symmetrical  figure, 
the  axis  being  the  perpendicular  that  bisects  it.  The  halves  of 
the  line  are  also  symmetrical  with  reference  to  the  midpoint  of 
the  line  as  a  center  of  symmetry. 

Corollary, — Two  figures,  or  parts  of  a  figure,  symmetrical 
with  reference  to  an  axis,  are  equal.  Every  case  of  demonstra- 
tion of  equality  by  rotation  on  an  axis  is  an  example  of  this  (III, 
10,  13,  and  26). 

6.  Theorem. — Every  diameter  bisects  the  circumference  and 
the  circle. 

If  the  part  on  one  side  of  the  diameter  is  turned  upon  that 
line  as  an  axis,  the  two  parts  coincide,  for  otherwise  some  points 
of  the  circumference  would  be  unequally  distant  from  the 
center. 

Corollaries. — I.  Every  circle  is  a  symmetrical  figure. 

II.  Any  diameter  of  a  circle  is  an  axis  of  symmetry. 

III.  Every  point  of  a  circumference  has  its  symmetrical 
point  with  reference  to  the  center,  as  a  center  of  symmetry. 

7.  Problem. — Arcs  of  equal  radii  may  he  added,  subtracted, 
multiplied,  or  divided. 

For  an  arc,  having  a  given  radius,  may  be  produced  to  any 
extent,  or  it  may  be  diminished  at  will. 

The  sum  of  several  arcs  may  be  greater  than  a  circumfer- 
ence. 

Scholium. — Two  arcs  not  having  the  same  radius  may  be 
joined  together,  and  the  result  may  be  called  their  sum  ;  but  it 
is  not  one  arc. 


Art.  1.]  CIRCUMFERENCES.  41 

The  circumference  is  the  only  line  that  can  move  along  itself, 
around  a  center.  For  any  line  that  can  do  this  must  have  all  its 
points  equally  distant  from  the  center  ;  that  is,  it  must  be  a  cir- 
cumference. 

8.  Applications. — The  axles  of  wheels,  shafts,  and  other  solid 
bodies  that  are  required  to  rotate  within  a  hollow  mold  or  casing  of  their 
own  form,  must  be  circular.  If  they  were  of  any  other  form,  they  could 
not  turn  without  carrying  the  mold  or  casing  with  them. 

Wheels,  intended  to  maintain  a  carriage  at  the  same  height  above 
the  road  on  which  they  roll,  must  be  circular,  with  the  axle  at  the 
center. 

Arcs  and  Chords. 

9.  Every  chord  divides  the  circumference  into  two  parts. 
Each  of  such  arcs  is  said  to  be  conjugate  to  the  other. 

A  Major  Arc  is  one  greater  than  a  semicircumference. 

A  Minor  Arc  is  one  less  than  a  semicircumference. 

In  speaking  of  the  arc  of  a  chord,  when  the  two  are  unequal, 
the  minor  arc  is  intended  unless  otherwise  expressed. 

A  chord  is  said  to  subtend  its  arc.  Any  arc  or  other  line 
extending  from  a  point  in  one  arm  of  an  angle  to  a  point  in  the 
other  arm  is  said  to  subtend  the  angle.  Thus,  any  side  of  a  tri- 
angle subtends  the  opposite  angle,  and  the  angle  made  by  two 
radii  is  subtended  by  an  arc. 

10.  Theorem. — If  two  arcs  are  equal,  their  chords  are 
equal. 

For  equal  arcs  may  coincide  ;  then  the  straight  lines  joining 
their  ends  must  coincide. 

11.  Theorem. —  When  a  radius  n 
bisects  an  arc,  it  is  per^yendicular  to                y""^ 
the  chord  of  the  arc  and  bisects  it.              / 

Let  CD  bisect  the  arc  AB.    The  / 

points  D  and  C  are  each  equally  dis- 
tant from  A  and  JB  (10).     There-  \ 
fore,  CD  is  perpendicular  to  AB  at  \ 
its  midpoint,  E  (III,  11,  c).  \^^ 
3 


42 


PLANE  GEOMETRY. 


[Chap.  IV. 


Corollaries. — I.  Since  two  conditions  determine  the  posi- 
tion of  a  straight  line,  if  it  has  any  two  of  the  four  con- 
ditions mentioned  in  the  theorem,  it  has  the  other  two.  These 
conditions  are  : 

1.  The  line  passes  through  the  center  of  the  circle,  that  is, 
it  is  a  radius. 

2.  It  passes  through  the  midpoint  of  the  chord. 

3.  It  passes  through  the  midpoint  of  the  arc. 

4.  It  is  perpendicular  to  the  chord. 

II.  The  angles  made  by  a  chord  with  the  radii  at  its  ends 
are  equal  (III,  10,  v). 


12.  Theorem. — Iii  the  same  circle,  or  in  equal  circles^  if 
two  minor  arcs  are  unequal,  the  greater  arc  has  the  greater 
chord. 

The  arc  AMB  being  greater 
than  CND,  both  being  minor  arcs, 
it  is  to  be  proved  that  AB  is  lon- 
ger than  CD. 

Take  AME,  equal  to  CND. 
Make  the  chords  AE  and  BE, 
and  the  radius  01,  perpendicular 
toJ5^. 

Since  the  arc  AMI  is  less  than 
a  semicircumference,  the  points  A 
and  E  are  on  the  same  side  of 
the   radius   that   is   perpendicular 

to  BE.     Therefore,  AB  is  longer  than  AE,  or  its  equal  CD 
(HI,  11). 

Corollaries. — I.  When  both  arcs  are  major  arcs,  the  greater 
arc  has  the  shorter  chord. 

II.  Conversely,  if  two  chords  are  unequal,  the  greater  chord 
has  a  greater  minor  arc  and  a  less  major  arc. 

IIL  When  the  chords  are  equal,  the  arcs  are  equal. 


Abt.  13.]  CIRCUMFERENCES.  43 

Distance  from  the  Center. 

13.  Theorem. — The  chords  of  equal  arcs  are  equally  dis- 
tant from  the  center. 

For  the  arcs,  being  equal,  may  coincide,  also  their  chords 
(10).  Then  the  perpendiculars,  which  measure  the  distance 
from  the  center,  must  coincide  (2). 

14.  Theorem, —  Of  two  unequal  chords,  in  the  same  circle 
or  in  equal  circles^  the  less  is  the  farther  from  the  center. 

Since  the  greater  chord,  AB,   has  a  greater  arc  than  the 
other,  GJDy  take  an  arc,  ^JF^  equal 
to  CD,  so  that   the   arcs  AB  and 
EF  have  the  same  midpoint  G. 

Then  the  radius  HG  is  perpen- 
dicular to  both  the  chords  AB  and 
EF  (11),  and  ^i^is  the  farther  from 
H,  because  its  distance  is  measured 
by  the  whole  of  SK,  and  the  dis- 
tance oi  AB  is  measured  by  a  part 
of  HK.  But  FF  and  CD  are 
equally  distant  from  the  center  (13). 

Corollary. — Conversely,  chords  at  equal  distances  from  the 
center  are  equal ;  and  of  two  chords  at  different  distances  from 
the  center,  the  farther  is  the  less. 

15.  Exercises, — 1.  What  is  the  locus  of  the  midpoints  of  all  the 
equal  chords  in  a  given  circle?  [Make  a  diagram  of  the  given  figure, 
designate  many  points  which  satisfy  the  condition,  describe  their  location, 
and  prove  that  the  points  in  this  locus  satisfy  the  condition  and  that 
no  other  points  do  so  (13  and  1,  iii).] 

2.  Find  the  locus  of  the  centers  of  the  circles  having  the  same  length 
of  radius,  whose  circumferences  pass  through  a  given  point.     [1,  iii.] 

3.  Find  the  locus  of  the  centers  of  the  circles  whose  circumferences 
pass  through  two  given  points.     [Ill,  11,  o.] 

4.  If  any  point,  not  the  center,  is  taken  in  a  diameter  of  a  circle,  of 
all  the  chords  that  pass  through  that  point,  that  is  the  least  which  is  at 
right  angles  to  the  diameter.  [Make  two  chords  through  the  point,  one 
perpendicular  and  the  other  oblique  to  the  diameter ;  and  consider  their 
respective  distances  from  the  center.] 


44 


PLANE  GEOMETRY. 


[Chap.  IV. 


Curvature. 

16.  Theorem. — A  straight  line  that  is  perpendicular  to  a 
radius  at  its  extremity  touches  the  circumference  in  only  one 
point. 

Let  AD  be  perpendicular  to  the 
radius  J30  at  its  extremity  J3.  Then 
it  is  to  be  proved  that  A  J)  touches 
the  circumference  at  no  other  point 
than  B. 

If  the  center  C  is  joined  by  a 
straight  line  with  any  point  of  AD, 
the  perpendicular  J3C  is  shorter  than 
any  such  oblique  line  (III,  10).  There- 
fore (1,  III),  every  point  of  the  line 
AD,  except  -S,  is  outside  of  the  cir- 
cumference. 

A  Tangent  is  a  line  touching  a  circumference  in  only  one 
point.  The  circumference  is  also  said  to  be  tangent  to  the 
straight  line.    The  common  point  is  called  the  point  of  contact. 


17.   Theorem. — A  straight  line  that  is  oblique  to  a  radius 
at  its  extremity  cuts  the  circumference  in  two  points. 

Let  AD  be  oblique  to  the  radius  CB  at  its  extremity  B, 

From  the  center  (7,  let  CJE  fall 
perpendicularly  on  AD.  On  ED, 
take  UF  equal  to  FB, 

The  distance  from  G  to  any  point 
of  the  line  AD  between  B  and  F 
is  less  than  the  length  of  the  radius 
GB,  and,  to  any  point  of  the  line 
beyond  B  and  F,  it  is  greater  than 
the  length  of  GB  (III,  10).  There- 
fore, that  part  of  the  line  between  B 
and  F  is  within,  and  the  parts  be- 
yond B  and  F  are  without  the  cir- 
cumference. That  is,  the  line  cuts  the  circumference  in  two 
points. 


Art.  17.]  CIRCU3IFERENCES.  \  47 

Corollaries. — I.  A  tangent  to  a  circumferenc\  % 

dius  that  extends  to  the  point  of  contact  are  perj 
each  other. 

II.  At  one  point  of  a  circumference,  there  can 
tangent  (III,  9). 

III.  The  two  directions  that  the  curve  has  at  the  point  of 
contact  are  the  same  as  those  of  the  tangent,  for  any  other 
straight  line  through  the  point  of  contact  cuts  the  curve. 

Whenever  a  curve  is  an  arm  of  an  angle,  the  direction  of 
that  arm  is  that  of  the  curve  at  the  vertex. 

A  Normal  is  a  line  perpendicular  to  a  curve.  In  the  circle 
every  radius  is  a  normal.  Sometimes  a  tangent  to  a  curve  is 
defined  as  a  straight  line  having  the  same  direction  as  the  curve 
at  anoint  that  is  common  to  the  two. 

18,  Theorem. — The  curvature  of  an  arc  of  a  circle  is  eqital 
to  that  angle  at  the  center  which  the  arc  subtends. 

The  curvature  of  an  arc  is  the  amount  of  change  in  direc- 
tion (II,  6).     Since  the  direction  of  the 
arc  AJ^  at  the  extremity  A  is  the  same  ^  CD 

as  that  of  the  tangent  AD,  and  the  direc- 
tion at  the  other  extremity  is  the  same  as 
that  of  the  tangent  (7-S,  the  curvature  of 
the  arc  is  the  angle  D  CJB  ;  but  this  an- 
gle is  equal  to  AEB  formed  by  the  radii 
(III,  25). 

Scholium. — The  amount  of  curvature 
being  intended,  an  arc  is  an  angular  quantity.    Thus  an  arc  is  said 
to  be  the  complement  of  an  angle,  when  the  sum  of  the  angle  and 
the  curvature  of  the  arc  is  one  right  angle,  or  ninety  degrees. 

As  the  subtending  arc  has  the  same  angular  quantity  as  the 
angle  at  the  center,  it  is  sometimes  said  that  "the  arc  meas- 
ures the   angle,"  but  this  is  not   a  proper  use   of  the  word 


19.  Application, — Tangent  lines  are  frequently  used  in  the  arts. 
A  common  example  is  when  a  strap  is  carried  round  a  part  of  the  cir- 
cumference of  a  wheel,  and  extends  to  a  distance. 


44 


PLANE  GEOMETRY. 
Angrles  at  the  Center. 


[Chap,  IV. 


30.  Theorem. — 171  the  same  circle^  or  in  equal  circles^  tico 
angles  at  the  center  have  the  same  ratio  as  the  arcs  that  subtend 
them. 

This  theorem  presents  three  cases  : 

1st.  If  the  arcs  are  equal. 


The  equal  arcs,  being  placed  one  upon  the  other,  coincide. 
Then  BG  coincides  with  AO,  and  DC  with  UO.  Thus  the 
angles  coincide  and  are  equal,  that  is,  they  have  the  same  ratio 
as  the  arcs. 

2d.  If  the  arcs  have  the  ratio  of  two  whole  numbers. 


Let  the  arcs  be  divided  into  equal  parts  corresponding  in 
number  to  the  terms  of  the  ratio  ;  these  small  arcs  are  all  equal. 
If,  for  example 

BB  :  AE  =  13  :  5, 

the  thirteenth  part  of  BB  is  equal  to  the  fifth  part  of  AE, 


Art.  20.]  CIRCUMFERENCES.  47 

Let  radii  be  made  to  all  the  points  of  division.  The  small 
angles  at  the  center  thus  formed  are  all  equal,  because  their  in- 
tercepted arcs  are  equal.  But  B  CD  is  the  sum  of  thirteen,  and 
A  OE  of  five  of  these  equal  angles.     Therefore, 

angle  BCD  :  angle  J.  0^=  13  :  5  ; 

that  is,  the  angles  have  the  same  ratio  as  the  arcs. 

3d.  The  arcs  may  have  a  ratio  that  can  not  be  expressed  by- 
two  whole  numbers,  that  is,  the  arcs  may  not  have  any  common 
measure.  It  is  to  be  proved  that  the  angles  have  still  the  same 
ratio  as  the  arcs,  that 

arc  BD  :  arc  AE=  angle  B  CD  :  angle  A  OE. 

If  this  proportion  is  not  true,  then,  the  first,  second,  and  third 
terms  being  unchanged,  the  fourth  term  is  either  too  large  or 


too  small.     If  it  were  too  large,  then  some  smaller  angle,  as 
A  07"  would  verify  the  proportion,  and 

arc  BD  :  arc  AE=  angle  BCD  :  angle  A  01. 

Let  the  arc  BD  be  divided  into  equal  parts,  each  of  them 
less  than  EI.  Let  one  of  these  parts  be  applied  to  the  arc  AE, 
beginning  at  A,  and  marking  the  points  of  division.  One  of 
those  points  must  fall  between  I  and  E^  say  at  the  point  XT. 
Join  OTI. 

Now,  by  construction,  the  arcs  BD  and  A  U  have  the  ratio 
of  two  whole  numbers.     Therefore, 

arc  BD  :  arc  AU=  angle  B  CD  :  angle  A  Oil. 


48 


PLANE  OEOMETRY, 


[Chap.  IY. 


These  two  proportions  have  the  same  antecedents  respective- 
ly.    It  follows  that  their  consequents  are  in  proportion  ;  that 


IS, 


arc  AE:  arc  AU=  angle  A  01  \  angle  AOIZ 


But  this  is  impossible,  for  the  first  antecedent  is  greater  than 
its  consequent,  while  the  second  antecedent  is  less  than  its  con- 
sequent.    Therefore,  the  supposition  that  led  to  this  conclusion 


is  false,  and  the  fourth  term  of  the  proportion,  first  stated,  is  not 
too  large.  It  may  be  shown,  in  the  same  way,  that  it  is  not  too 
small. 

Therefore,  the  angle  A  OE  is  the  true  fourth  term  of  the 
proportion,  that  is,  the  arc  BD  is  to  the  arc  AE  as  the  angle 
BCD  is  to  the  angle  A  OE, 

Corollaries. — I.  Any  angle  at  the  center  has  the  same  ratio 
to  the  sum  of  four  right  angles  that  the  intercepted  arc  has  to 
the  whole  circumference. 

II.  If  two  diameters  are  perpendicular  to  each  other,  they 
divide  the  circumference  into  four  equal  parts  ;  and  conversely, 

III.  The  curvature  of  an  arc  of  a 
circle  increases  in  the  same  ratio  as  its 
linear  extent.  That  is,  the  curvature 
of  a  circumference  of  a  circle  is  uniform 
throughout. 

A  Quadrant  is  the  fourth  part  of 
a  circumference.  It  is  divided  into  de- 
grees, minutes,  and  seconds,  the  same 
as  a  right  angle. 


Art.  21.] 


CIRCUMFERENCES. 


49 


31.  Exercises. — 1.  "What  is  the  locus  of  the  midpoints  of  all  the 
chords  that  are  parallel  to  a  given  tangent  ? 

2.  If  two  circles  are  concentric,  a  line  that  is  a  chord  of  the  outer 
and  a  tangent  of  the  inner  one  is  bisected  at  the  point  of  contact.  [Make 
a  radius  through  the  point  of  contact.] 

3.  If  two  concentric  circles  are  cut  bj  the  same  secant,  the  segments 
of  the  secant  between  the  circumferences  are  equal.  [Make  a  radius  per- 
pendicular to  the  secant.] 

4.  If  the  length  of  a  circumference  is  one  meter,  what  is  the  length 
of  the  arc  that  subtends  an  angle  at  the  center,  of  7°  12'? 

5.  If  the  arc  subtending  a  central  angle  of  25"  is  one  inch  in  length, 
how  long  is  the  whole  circumference  ? 

6.  If  a  circumference  is  3|  times  as  long  as  its  diameter,  is  an  arc  of 
58°  48'  longer  or  shorter  than  the  radius  ? 


Angles  and  Parallels  in  a  Circumference. 

22.  An  Inscribed  Angle  is  one  formed  by  chords,  with 
its  vertex  on  the  circumference. 

An  inscribed  angle  is  said  to  stand 
upon  the  arc  that  subtends  it ;  the  re- 
mainder of  the  circumference  is  said 
to  contain  the  angle ;  and  the  an- 
gle is  in  the  arc  that  contains  it. 
Thus  the  angle  AEI  stands  upon  the 
arc  A  01,  and  is  in  the  arc  AEI, 
■which  contains  it. 

Theorem. — An  inscribed  angle  is  equal  to  the  curvature  of 
half  the  arc  it  stands  upon. 

The  demonstration  presents  two  cases. 

1st.  When  one  arm  of  the  angle  is  a  diameter,  as  JBA.     It 
is  to  be  proved  that  the  angle  JB  is 
equal  to  the  curvature  of  half  the  arc 
AEG. 

Make  the  radius  DE  parallel  to 
BC,  and  join  CD. 

The  angles  B  and  C  are  equal  (11, 
ii)  ;  B  is  equal  to  its  corresponding 
angle  ABE  (III,  22)  ;  and  G  is  equal 
to  its  alternate  angle  GBE  (III,  22, 


50 


PLANE  GEOMETRY. 


[Chap.  IV. 


ii).  Therefore,  the  arcs  AE  and  JEG 
are  equal  (20),  and  the  angle  B  is 
equal  to  the  curvature  of  the  arc  AE, 
which  is  half  of  the  subtending  arc 
AEC. 

2d.  When  neither  arm  of  the  an- 
gle is  a  diameter.  Make  the  diam- 
eter BD. 

The  angle  ABB  is  equal  to  the 
curvature  of  half  the  arc  AB,  as  just  proved,  and  the  angle 
BBC  is  equal  to  the  curvature  of  half  the  arc  BG.     There- 


fore, taking  the  sum  or  difference  according  as  the  center  of 
the  circle  is  between  the  arms  of  the  angle  or  not,  the  angle 
AB  G  is  equal  to  the  curvature  of  half  the  arc  A  G. 

Corollaries, — I.  When  an  inscribed  angle  and  an  angle  at 
the  center  have  the  same  subtending  arc,  the  inscribed  angle  is 
half  of  the  angle  at  the  center. 

II.  Several  inscribed  angles,  in  the 
same  arc  or  equal  arcs,  are  equal ;  and 
conversely,  if  inscribed  angles  are 
equal  and  in  the  same  circle  or  in 
equal  circles,  then  their  subtending 
arcs  are  equal. 

III.  An  inscribed  angle  is  obtuse, 
right,  or  acute,  according  as  the  arc 
that  subtends  it  is  greater  than,  equal 
to,  or  less  than  a  semicircumference. 


Art.  22.] 


CIRCUMFERENCES. 


51 


lY.  The  angles  inscribed  in  conjugate  arcs  are  supplemen- 
tary, for  their  sum  is  equal  to  the  curvature  of  half  the  circum- 
ference. 


33.  Theorem, —  The  angle  formed  by  a  tangent  and  a  chord 
at  the  point  of  contact  is  equal  to  the  curvature  of  half  of  the 
subtending  arc. 

Making  the  diameter  J^D  from 
the  point  of  contact,  DEC  is  a 
right  angle,  and  therefore  equal 
to  the  curvature  of  half  the  semi- 
circumference  DIOEy  but  the  in- 
scribed angle  DEI  is  equal  to  the 
curvature  of  half  the  arc  DL  Sub- 
tracting this  angle  and  arc  from 
the  right  angle  and  semicircum- 
f  erence,  the  remaining  angle  lEG 
is  equal  to  the  curvature  of  half 
the  arc  lOE. 

By  addition,  the  angle  lEA  is  equal  to  the  curvatur#of  half 
the  subtending  arc  IDE. 


24.  Theorem. — Two  parallel  lines  intercept  equal  arcs  of  a 
circle. 


4     n 


For,  joining  B  Cy  the  alternate  angles  AB  G  and  B  CD  are 
equal.     Therefore,  the  subtending  arcs  are  equal. 


52 


PLANE  GEOMETRY. 


[Chap.  IY. 


25.  Theorem. — Every  angle  lohose  vertex  is  within  the 
circumference  is  equal  to  the 
curvature  of  half  the  sum  of 
the  arcs  intercepted  between  its 
arms  and  between  its  arms 
produced. 

Thus,  the  angle  Z/AE  is 
equal  to  the  curvature  of  half 
the  sum  of  the  arcs  UJE  and 
10. 

To  be  demonstrated  by 
the  student,  using  the  previous  theorems  (22  and  24). 


26.  Theorem. — Every  angle  whose  vertex  is  outside  of  a  cir- 
cumference, and  whose  arms  are  either  tangent  or  secant,  is  equal 
to  the  curvature  of  half  the  difference  of  the  subtending  arcs. 

Thus,  the  angle  A  CF  is 
equal  to  the  curvature  of  half 
the  difference  of  the  arcs  AF 
and  AB ;  the  angle  FOG,  to 
that  of  half  the  difference  of 
the  arcs  FGr  and  BI',  and 
the  angle  ACE,  to  that  of 
half  the  difference  of  the  arcs 
AFGE  and  ABIE. 

This,  also,  may  be  demon- 
strated by  the  student. 

Corollaries. — I.  Accord- 
ing as  an  angle  subtended  by  a  chord  has  its  vertex  outside, 
on,  or  inside  the  arc,  it  is  less  than,  equal  to,  or  greater  than  the 
angle  contained  by  the  arc. 

II.  Conversely,  according  as  an  angle  that  is  subtended  by  a 
chord  is  less  than,  equal  to,  or  greater  than  the  angle  contained 
by  the  arc  of  that  chord,  its  vertex  is  outside,  on,  or  within  the 
arc. 

III.  The  locus  of  the  vertices  of  all  angles  that  are  equal  to 
a  given  angle  and  are  subtended  by  a  given  line  is  an  arc  of  a 
circle. 


Art.  26.]  CIRCUMFERENCES.  53 

IV.  If  two  angles  are  supplementary  and  have  their  vertices 
on  opposite  sides  of  a  common  subtending  line,  a  circumfer- 
ence may  pass  through  the  vertices  and  the  ends  of  the  line 
(22,  IV). 

Scholium. — All  the  cases  of  arcs  between  two  lines  that  cut 
or  touch  a  circumference  are  included  in  this  general  rule  :  The 
angle  formed  by  the  lines  is  equal  to  the  curvature  of  half  of  the 
arc  or  sum  of  the  arcs  concave  to  the  vertex,  less  half  that  arc, 
if  any,  which  is  convex  to  the  vertex. 

The  case  of  arcs  between  parallels  is  included  in  this  rule,  for 
there  is  no  angle,  no  difference  of  direction  of  the  two  lines,  and 
there  is  no  difference  between  the  intercepted  arcs. 


2i*7  •  Applications. — Instruments  for  measuring  angles,  founded 
upon  the  principle  that  arcs  are  proportional  to  angles,  consist  of  a  part  or 
an  entire  circle  of  metal,  on  which  are  engraved  its  divisions  into  degrees, 
etc.  Many  instruments  used  by  surveyors,  navigators,  and  astronomers, 
are  constructed  upon  this  principle. 

An  instrument  called  a  protractor  is  used  in  drawing  angles,  or  meas- 
uring angles  in  a  drawing.     It  consists  of  a  semicircle,  the  arc  of  which 
is  divided  into  degrees  and  parts  of  a 
degree. 

A    method  of   surveying  a  railway  ^  \ 

curve  depends  upon  the  principles  just  \ 

established.  ^r'''-^-"*"""^H/\ 

Suppose  that  AB  and  CD  are  straight  /y'^""'""^^^^-^<\ 

parts  of  the  track  arid  are  to  be  connected  '^/"  "V' 

by  a  curve,  the  points  B  and  B  being  so  /  \ 

selected  that  the  angles  ABB  and  BBC         /A  (A 

are    equal.      Two    transits   (instruments 
for  measuring  angles)  are  used,  one   at 

B  pointed  toward  B,  the  other  at  B  pointed  toward  E  in  the  con- 
tinuation of  CB.  If  the  first  transit  is  turned  toward  E  and  the 
second  toward  B,  both  the  same  number  of  degrees,  the  point  of  in- 
tersection of  their  lines  of  sight  is  on  the  curve.  Any  number  of 
such  points  may  be  ascertained  without  moving  the  instruments  from  B 
and  B. 


54 


PLANE  GEOMETRY. 


[Chap.  IV. 


28.  Exercises. — 1.  The  opposite  lines  joining  the  ends  of  two 
diameters  in  a  circle  are  parallel.     [Ill,  7,  ix ;  IV,  22,  i ;  III,  24,  i.] 

2.  If  two  equal  circumferences 
cut  each  other  at  the  points  A  and 
J5,  and  any  line  is  made  through  B 
to  cut  the  curves  at  G  and  at  2), 
then  the  chord  AG  \^  equal  to  the 
chord  AD.     [Join  AB.] 

3.  If  any  two  circumferences 
cut  each  other  at  the  points  A  and 
J?,  and  a  tangent  is  made  to  each 
curve  at  A  and  extended  to  the 
other  curve  at  G  and  at  i>,  then  the 
angles  GBA  and  DBA  are  equal. 
[Produce  DA  and  GA  beyond  Ai] 

4.  If  from  one  point  there  ex- 
tend two  lines  tangent  to  a  circum- 
ference, the  angle  contained  by  the 
tangents  is  double  the  angle  contained 
by  the  line  joining  the  points  of  con- 
tact and  the  radius  extending  to  one 
of  them.    [There  are  several  ways  to 

demonstrate  this.  The  construction  for  one  way  is :  Make  two  diameters 
from  the  points  of  contact.  For  another  way  :  From  one  of  the  points  of 
contact  make  a  chord  parallel  to  the  other  tangent,  and  from  the  other 
point  of  contact  make  a  diameter.] 

5.  Find  the  locus  of  the  midpoints  of  all  the  chords  in  a  circle 
which  extend  from  the  same  point  on  the  circumference.  [11,  i,  and 
26,  III.] 


Positions  of  two  Circumferences. 


29.  The  various  relative  positions  of  two  circles  depend 
upon  the  distance  between  their  centers  compared  with  the  siim 
or  the  difference  of  their  radii.     This  is  shown  as  follows  : 

Since  every  diameter  is  an  axis  of  symmetry  of  a  circle,  the 
line  joining  the  centers  of  two  circles  is  an  axis  of  symmetry  of 
the  figure.     This  line  is  called  the  central  line. 

A  perpendicular  to  the  central  line,  at  a  point  where 
one  of  the  curves  cuts  it,  must  be  tangent  to  that  curve 
(16). 


Art.  29.] 


CIRCUMFERENCES. 


55 


If  both  curves  cut  the  central  line  at  the  same  point,  the  per- 
pendicular at  that  point  is  a  common  tangent,  and  the  circum- 
ferences are  tangent  to  each  other. 

Corollaries. — I.  When  two  circumferences  are  exterior  to 
each  other,  the  distance  between  their  centers  is  greater  than 
the  sum  of  their  radii. 


\ 


h 


II.  "When  one  of  two  circumferences  is  within  the  other,  the 
distance  between  the  centers  is  less  than  the  difference  of  the 
radii. 

III.  When  they  touch  each  other  exteriorly  the  distance  be- 
tween the  centers  is  equal  to  the  sum  of  the  radii. 


lY.  When  they  touch  each  other  interiorly,  the  distance  be 
tween  the  centers  is  equal  to  the  difference  of  the  radii. 

Y.  When  they  cut  each  other, 
the  distance  between  the  centers  is 
less  than  the  sum,  but  greater  than 
the  difference  of  the  radii. 

YI.  The  above  five  cases  are  all 
that  are  possible.  Therefore  the 
converse  of  each  proposition  is 
true. 


56  PLANE  GEOMETRY.  [Chap,  IV. 

VII.  The  common  chord  of  two  intersecting  circumferences 
is  perpendicular  to  the  central  line  and  is  bisected  by  it  (III, 
11,  c). 

VIII.  Two  concentric  circumferences  are  equidistant  at  all 
points. 

30.  Exercises. — 1.  If  the  radii  of  two  circles  are  67  and  78  milli- 
meters, and  they  are  placed  with  their  centers  140  millimeters  apart,  do 
the  curves  cut  or  touch  each  other  ?  If  the  centers  are  145  millimeters 
apart?     If  150? 

2.  How  is  it  when  the  centers  are  10  millimeters  apart?  "When  11? 
When  12? 

3.  What  is  the  locus  of  the  centers  of  those  circles  whose  circumfer- 
ences touch  a  given  line  at  a  given  point? 

4.  Of  the  centers  of  those  which  touch  a  given  arc  at  a  given  point  ? 

5.  When  two  circumferences  have  no  common  point,  the  least  dis- 
tance between  the  curves  is  measured  along  the  central  line. 

6.  On  any  two  circumferences,  the  two  points  which  are  at  the  great- 
est distance  apart  are  in  the  central  line. 

7.  In  each  of  the  five  cases  of  Article  29,  how  many  straight  lines  can 
be  tangent  to  both  circumferences?  [The  number  is  different  for  each 
case.] 

8.  If  two  circumferences  are  tangent  to  each  other,  and  two  secants 
are  made  through  the  point  of  contact,  the  chords  which  Join  the  ends  of 
these  secants  are  parallel.  [At  the  point  of  contact,  make  a  straight  line 
tangent  to  the  curves,  and  compare  angles.] 

9.  Can  two  chords  of  the  same  circle,  that  are  not  both  diameters, 
bisect  each  other?     [Ill,  9.] 

10.  A  circle  of  one  inch  radius  being  given,  find  the  locus  of  the 
centers  of  the  circles  of  three  inches  radius,  all  of  which  are  tangent  to 
the  given  circle. 

31.  Scholia, — The  study  of  the  circumference  has  served 
to  develop  the  notion  of  symmetry  ;  also  to  show  the  close  con- 
nection between  angles  and  arcs  of  circles. 

This  chapter  completes  the  study  of  plane  figures  that  consist 
in  lines  without  reference  to  inclosed  surface. 

Elementary  Geometry  treats  only  of  figures  every  line  of 
which  is  either  straight  or  of  uniform  curvature.  Higher  Geom- 
etry treats  of  curves  of  varying  curvature. 


Art.  31.]  CIRCUMFERENCES.  57 

Modes  of  Geometrical  Reasoning, 

I.  A  Direct  Demonstration  proceeds  from  established 
premises  by  a  regular  deduction. 

An  Indirect  Demonstration  begins  with  the  conclusion. 
It  proceeds  by  these  steps  : 

1st.  Suppose  that  the  conclusion  to  be  demonstrated  is  not 
true.     This  supposition  is  called  the  false  hypothesis. 

2d.  Show,  by  reasoning  upon  the  false  hypothesis,  that  it 
involves  a  contradiction,  or  leads  to  an  impossible  conclusion. 
This  contradiction  or  impossibility  is  called  the  absurd  conclu- 
sion, and,  hence,  this  method  is  called  reductio  ad  absnr- 
dum. 

3d.  Since  the  supposition  that  the  conclusion  is  false  leads  to 
an  absurdity,  the  conclusion  must  be  true. 

For  example,  take  Article  11  of  Chapter  II.  The  false  hypo- 
thesis is,  that  two  planes  that  coincide  to  some  extent  may 
diverge.  It  would  follow  from  this  that  some  of  the  directions 
of  at  least  one  of  them  may  vary.  As  this  contradicts  the  defi- 
nition of  a  plane  it  is  absurd. 

This  method  of  argument  is  as  common  in  other  matters  as 
in  mathematics. 

Some  modes  of  reasoning,  such  as  the  method  of  super- 
position (II,  4,  and  III,  31),  are  peculiar  to  the  science  of 
Magnitude. 

II.  The  Method  of  Exhaustions  was  used  by  Euclid  in 
the  demonstration  of  a  theorem  that  involved  a  ratio  between 
magnitudes  that  have  no  common  measure.  For  an  example  of 
this  method,  see  the  third  case  of  Article  20.  It  involves  a 
double  application  of  the  reductio  ad  ahsurdum. 

In  modern  times,  various  attempts  have  been  made  to  substi- 
tute some  demonstration  equally  logical  and  less  tedious.  This 
has  not  been  done  without  a  substantial  adherence  to  the  mode 
of  thought  pursued  by  Euclid. 

III.  The  IVCethod  of  Limits  is  one  of  the  substitutes  for 
the  Method  of  Exhaustions.  It  is  used  in  the  demonstration  of 
the  theorem  in  Article  28,  Chapter  III.  The  use  of  this  method 
has  generally  been  attended  with  fallacies.     The  most  common 


58  PLANE  GEOMETRY.  [Chap.  IV. 

error  is  the  neglect  to  prove  that  the  constant  quantity  is  the 
limit  of  the  variable.  It  has  also  been  assumed,  without  proof, 
that  any  rule  of  measurement  which  applies  to  a  variable  magni- 
tude holds  true  of  its  limit.  Dr.  Whewell,  of  Trinity  College, 
Cambridge,  claimed  this  as  axiomatic,  but  whatever  is  true  in 
the  statement  can  be  demonstrated  with  as  much  rigor  as  is 
found  in  the  ancient  method  of  exhaustions. 

Theorem. — "If  two  variables  are  equal  at  every  step  of  their 
variation  and  if  each  has  a  limit,  the  limits  are  equals 

The  following  demonstration  is  substantially  the  same  as 
that  of  Duhamel,  who  first  stated  this  theorem. 

If  the  limits  were  unequal,  that  is,  if  they  differed  by  any 
quantity,  then  one  of  the  variables  might  pass  over  the  limit  of 
the  other  variable,  in  order  to  reach  within  less  than  this  quan- 
tity of  difference  from  its  own  limit.  Then  one  of  the  two  varia- 
bles would  be  greater  and  the  other  less  than  the  limit  passed  ; 
which  is  absurd,  as  they  remain  equal  by  hypothesis.  Therefore 
the  limits  can  not  be  unequal. 

By  the  Euclidean  method  of  exhaustions,  the  magnitude  ex- 
hausted was  a  limit.  An  essential  part  of  the  demonstration 
consisted  in  showing  that  this  magnitude  could  be  exhausted 
within  less  than  any  assignable  quantity. 

lY.  The  Method  of  Indivisibles  consists  in  regarding 
magnitudes  as  composed  of  infinitely  small  elements,  called 
Infinitesimals  or  Indivisibles.  "  A  line  is  said  to  consist  of 
points,  a  surface  of  parallel  lines,  and  a  solid  of  parallel  sur- 
faces." 

Two  equal  lines  consist  of  an  equal  number  of  points,  for  the 
two  lines  may  coincide  so  that  every  point  in  each  has  its  corre- 
sponding point  in  the  other.  It  follows  that  if  one  line  is  twice 
as  long  as  another,  there  are  twice  as  many  points  in  the  first  as 
in  the  second  ;  and  universally,  the  numbers  of  the  points  in  two 
lines  have  the  same  ratio  as  the  lengths  of  the  lines.  The  num- 
bers are  infinite,  but  their  ratio  is  finite. 

Apply  this  to  the  discussion  of  angles  at  the  center  and  their 
intercepted  arcs.  Radii  are  supposed  to  extend  from  every 
point  of  the  two  arcs.  These  radii  make  equal  infinitesimal  an- 
gles at  the  center,  one  such  angle  for  every  point  of  the  arc.     It 


Art.  81.]  CIRCUMFERENCES.  59 

follows  that  the  whole  angles  have  the  same  ratio  as  the  whole 
arcs,  as  in  the  second  case  in  the  demonstration  in  Article  20. 

These  methods  are  an  economy,  not  merely  of  words.  They 
have  saved  much  intellectual  labor,  and  contributed  largely  to 
the  progress  of  mathematical  science. 


32.  Miscellaneous  Exercises, — 1.  If  from  a  point  without  a 
circle  two  straight  lines  extend  to  the  concave  part  of  the  circumference, 
making  equal  angles  with  the  line  joining  the  same  point  and  the  center 
of  the  circle,  then  the  parts  of  the  first  two  lines  witiiin  the  circumfer- 
ence are  equal.     [Ill,  13.] 

2.  What  is  the  locus  of  the  centers  of  those  circles  that  have  a  radius 
of  the  same  given  length,  and  which  are  tangent  to  the  same  given 
circle?     [Generalize  Exercise  10,  Article  30.] 

3.  If  two  circumferences  are  such  that  the  radius  of  one  is  the  diam- 
eter of  the  other,  any  straight  line  extending  from  their  point  of  contact 
to  the  outer  circumference  is  bisected  by  the  inner  one.  [Join  the  center 
of  the  larger  circle  and  the  point  where  the  straight  line  cuts  the  smaller 
circumference.] 

4.  If  two  circumferences  cut  each  other,  and  from  either  point  of  in- 
tersection a  diameter  is  made  in  each,  the  extremities  of  these  diameters 
and  the  other  point  of  intersection  are  in  the  same  straight  line.  [Make 
the  common  chord.] 

6.  In  a  given  circle  find  the  locus  of  the  midpoints  of  the  chords  that 
pass  through  a  given  point  within  the  curve. 

6.  If,  from  the  ends  of  a  diameter,  perpendiculars  fall  on  any  straight 
line  that  cuts  the  circumference,  the  segments  of  the  lino  intercepted  be- 
tween those  perpendiculars  and  the  curve  are  equal. 


Problems  in  Drawing. 

33.  The  solution  of  Problems  in  Drawing  is  a  test  of  a 
student's  knowledge  of  geometrical  principles  and  an  exercise  of 
his  skill  in  their  application. 

Except  the  pencil  or  crayon,  the  only  instruments  used  are 
the  ruler  and  compasses.  The  ruler  has  a  straight  edge. 
The  compasses  have  two  legs  with  pointed  ends,  which  meet 
when  the  instrument  is  shut.  For  blackboard  work,  a  stretched 
cord  may  be  substituted  for  the  compasses.     The  ruler  is  used  to 


60  PLANE  GEOMETRY.  [Chap.  IV. 

draw  straight  lines.  The  compasses  are  used  to  draw  circum- 
ferences, or  arcs  of  circles. 

That  this  much  should  be  taken  for  granted  was  expressed 
by  Euclid  in  these  three  postulates  : 

Postulates  of  Euclid. — 1.  A  straight  line  can  he  drawn 
from  any  point  to  any  point. 

2.  A  given  straight  line  can  he  produced  any  length,  in  the 
same  direction. 

3.  A  circumference  can  he  descrihed  with  any  center  and  with 
any  radius. 

Since  the  straight  line  and  the  circumference  are  the  only- 
lines  treated  of  in  elementary  geometry,  these  Euclidian  Postu- 
lates contain  all  that  need  be  granted  for  the  solution  of  elemen- 
tary problems  in  drawing.  The  rule  forbidding  the  use  of  any 
instruments  except  the  ruler  and  compasses  is,  in  effect,  a  restric- 
tion to  the  use  of  these  elementary  lines. 

By  means  of  the  compasses  a  part  of  a  line  may  be  taken 
equal  to  a  given  line. 

Problems  in  geometry  are  distinct  from  problems  in  drawing. 
The  former  are  pure  mathematical  principles  ;  the  latter  are  ap- 
plications of  principles  to  handiwork.  The  former  are  state- 
ments that  certain  magnitudes  can  exist.  Their  truth  is  made 
manifest  by  an  explanation  which  shows  that  the  definition  or 
description  of  the  figure  is  not  incompatible  with  itself.  Prob- 
lems in  drawing  are  tasks  that  are  to  be  done  under  certain  con- 
ditions. 

In  the  Elements  of  Euclid,  which,  for  many  ages,  was  the 
only  text-book  on  Elementary  Geometry,  the  problems  in  draw- 
ing occupy  the  place  of  problems  in  geometry.  At  present 
nearly  all  mathematicians  put  them  aside  as  not  forming  a  neces- 
sary part  of  the  theory  of  the  science. 

The  Postulates  of  Euclid  express  no  complete  scientific  truth. 
They  are  only  a  partial  statement  of  elementary  possibilities. 
Therefore  it  is  an  erroneous  theory  that  makes  them  the  basis  of 
any  demonstration  of  principle. 

The  student  is  advised  to  make  a  drawing  of  every  problem. 
First  draw  the  parts  given,  then  the  construction  requisite  for  solu- 
tion.   Endeavor  to  make  the  drawing  as  exact  as  possible.    Let  the 


Art.  33.]  CIRCUMFERENCES.  61 

lines  be  fine  and  even,  as  they  better  represent  the  abstract  lines 
of  geometry.  Accuracy  in  drawing,  like  precision  in  language, 
is  an  aid  to  correct  thought. 

Problems. — I.   To  find,  the  least  common  multiple  of  two  straigJit 
lines. 

Take  for  example  the  lines  ^  g  ^ 

and  c. 

From  a  point,  A^  draw  an  in- 
definite straight  line,  AE.    Apply  each  of  the  given  lines  to  it  a  number 


A ' r-^ ' ^ ' i-i ' ^ 


of  times  in  succession.  If  the  ends  coincide  for  the  first  time  at  E^  then 
AE  \^  the  least  common  multiple  of  the  two  lines. 

If  two  incommensurable  lines  are  given,  as  those  in  Article  2,  Chap- 
ter III,  theoreticallj  the  coincidence  is  never  reached,  but  apparently  a 
coincidence  can  be  obtained,  and  such  a  result  gives  apparently  the  least 
common  multiple.  This  exercise  and  the  following  may  be  made  tests  of 
accuracy  in  drawing. 

II. — T'o  find  the  greatest  common  measure  of  two  straight  lines. 

Subtract  the  smaller  line  from  the  greater  as  many  times  as  it  can  be 
taken.  Subtract  the  remainder,  if  any,  from  the  smaller  line,  in  the  same 
way.  Then  use  the  remainder  as  a  subtrahend,  and  so  on,  until  after 
some  subtraction  there  is  no  remainder.  The  last  subtrahend  is  the 
greatest  common  measure. 

Compare  this  with  the  rule  for  numbers,  or  algebraic  quantities.  Also 
compare  the  demonstration  in  Article  2,  Chapter  III. 

III. — To  find  the  ratio  of  two  straight  lines. 

Count  how  many  times  each  is  contained  in  the  least  common  multi- 
ple.    These  numbers  express  the  ratio  inversely. 

Or,  having  found  the  greatest  common  measure  of  the  lines,  reverse 
the  steps  and  count  how  many  times  the  common  measure  is  contained  in 
each.  Both  these  methods  are  liable  to  all  the  sources  of  error  that  arise 
from  frequent  measurements.  In  practice,  it  is  usual  to  measure  each 
line  as  nearly  as  may  be  with  a  comparatively  small  standard.  The  num- 
bers thus  found  express  the  ratio  nearly. 

Whenever  two  lines  have  any  geometrical  dependence  upon  each 
other,  the  ratio  is  found  by  calculation  with  an  accuracy  no  measurement 
by  the  hand  can  reach. 


62  PLANE  GEOMETRY.  [Chap.  IV. 

lY. — To  lisect  a  given  straight  line. 

With  A  and  JB  as  centers,  and  V2) 

with  the  same  radius,  which  must  be  • 

greater  than  the  half  of  AB,  describe 
arcs  which  intersect  at  D.  In  the 
same  way  describe  two  arcs  intersect- 
ing  at  some  other  point,  as  E.    Join  j 

DE.  I 

The  line  DE  has  two  points  each  j/ 

equally  distant  from  the  ends  of  AB,  ^^^ 

and  therefore  bisects  it. 

Y. — To  erect  a  perpendicular  on  a  given  line. 

To  erect  at  C  a  perpendicular  to  AB^  take  CB  and  GA  equal.  De- 
scribe, as  in  the  last  problem,  two  arcs  intersecting  at  i),  and  draw  DC. 
It  is  perpendicular. 

The  demonstration  of  this  and  of  some  that  follow  is  left  to  the  student. 

YI. — To  let  fall  a  perpendicular  from  a  given  point  on  a  given  straight 
line. 

With  the  given  point  as  a  center, 
and  a  radius  long  enough,  describe  j     i 

an  arc  cutting  the  given  line  BG  in  j 

the  points  D  and  E.     With  D  and  E  i 

as  centers,  and  with  a  radius  greater  j 


Ey_c 


than  the  half  of  DE^  describe  arcs        jj       n^j9 
cutting  each  other  in  F.   The  straight  ""~-~ 

line  joining  A  and  F  is  perpendicular 
to  DE.  ^ 

YIL— To  describe  a  circumference 
through  three  given  points. 

The  solution  is  evident  from  Article  2. 

YIII. — To  find  the  center  of  a  circle  when  all  or  a  part  of  the  circum- 
ference is  given. 

Take  any  three  points  of  the  arc  and  proceed  as  in  the  last  problem. 

IX. —  To  draw  a  tangent  at  a  given  point  of  an  arc. 

Draw  a  radius  to  the  given  point,  and  erect  a  perpendicular  to  the 
radius  at  that  point. 

X. —  To  Used  a  given  arc. 

Draw  the  chord  of  the  arc  and  erect  a  perpendicular  at  its  mid- 
point. 

Corollary.— To  bisect  an  angle,  first  draw  an  arc  with  the  vertex  as 
center. 


Art.  33.] 


CIRCUMFERENCES, 


"  The  most  simple  case  of  the  division  of  an  arc,  after  its  bisectiol,  .o 
its  trisection,  or  its  division  into  three  equal  parts.  This  problem  accord- 
ingly exercised,  at  an  early  epoch  in  the  progress  of  geometrical  science, 
the  ingenuity  of  mathematicians,  and  has  become  memorable  in  the  his- 
tory of  geometrical  discovery,  for  having  baffled  the  skill  of  the  most 
illustrious  geometers. 

"Its  object  was  to  determine  meaos  of  dividing  any  given  arc  into 
three  equal  parts,  without  any  other  instruments  than  the  rule  and  com- 
passes permitted  by  the  postulates  prefixed  to  Euclid's  Elements.  Simple 
as  the  problem  appears  to  be,  it  never  has  been  solved,  and  probably 
never  will  be,  under  the  above  conditions." — Lardner'%  Treatise. 

It  must  not  be  inferred  that  there  is  any  impossibility  in  the  trisection 
of  an  angle.  Greek  geometers  solved  the  problem  with  other  instru- 
ments, by  which  they  drew  curves  that  have  not  a  uniform  curvature. 

XI. — To  draw  an  angle  equal  to  a  given  angle. 

Let  it  be  required  to  draw  a  line 
making,  with  the  line  BG^  an  angle  at 
B  equal  to  the  angle'  A. 

With  ^  as  a  center,  and  any  as- 
sumed radius,  AD^  draw  the  arc  DE 
cutting  the  arms  of  the  angle  A.  With 
jB  as  a  center,  and  the  same  radius  as 
before,  draw  an  arc,  FG.  With  i^  as  a 
center,  and  a  radius  equal  to  the  chord 
BE^  draw  an  arc  cutting  FG  at  the 
point  G.  Join  BG.  Then  GBF  is  the 
required  angle. 

For  the  arcs  DE  and  FG  have  equal 
radii  and  equal  chords,  and  therefore  are  equal.     Hence,  they  subtend 
equal  angles. 

Corollary, — An  arc  equal  to  a 
given  arc  may  be  drawn  in  the  same 
way. 

XII. —  To  draw  an  angle  equal  to 
the  sum  of  two  given  angles. 

Let  A  and  B  be  the  given  angles. 
First,  make  the  angle  DOE  equal  to  A^ 
and  then  at  C,  on  the  line  CE^  draw 
the  angle  EGF  equal  to  B.  The  an- 
gle FGD  is  equal  to  the  sum  of  A 
and  5. 


64 


PLANE  GEOMETRY. 


[Chap.  IV. 


XIII. — To  erect  a  perpendicular  to  a  given  line  at  its  extreme  pointy 
without  producing  ilie  line. 

A  right  angle  may  be  made  separately,  and  then,  at  the  end  of  the 
given  line,  an  angle  equal  to  the  right  angle. 

This  is  the  method  employed  by  mechanics  and  draughtsmen 
to  construct  right  angles  and  perpendiculars  by  the  use  of  the 
square. 

XIV. — To  draw  a  line  tJirough  a  given  point  parallel  to  a  given  line. 

Draw  a  line  from  the  given  point  to  any  point  of  the  given  line. 
Then  through  the  given  point  draw  a  line  making  the  alternate  angles 
equal. 

XV. — Through  a  given  point  out  of  a  circumference^  to  draw  a  tan- 
gent to  the  circumference. 

Let  A  be  the  given  point,  and  G  the  center  of  the  given  circle.  Join 
AG.  Bisect  AG  at  the  point 
£.  "With  5  as  a  center  and 
£G  as  a  radius,  describe  a  cir- 
cumference. It  cuts  the  given 
circumference  in  two  points, 
JD  and  E.  Draw  straight  lines 
from  A  through  I)  and  E. 
AD  and  AE  are  tangents  to 
the  given  circumference. 

Join  GD  and  GE.  The  an- 
gle GBA,  inscribed  in  a  semi- 
circumference,   is   therefore  a 

right  angle.     AD,  perpendicular  to  the  radius  GD,  is  tangent  to  the  cir- 
cumference. 

XVI. —  Upon  a  given  chord 
to  describe  an  arc  that  contains 
a  given  angle. 

Let  AB  be  the  chord,  and 
G  the  angle.  Make  the  angle 
DAB  equal  to  G.  At  A  erect 
a  perpendicular  to  AD,  and 
erect  a  perpendicular  to  ^^  at 
its  midpoint.  Produce  these 
till  they  meet  at  F  (III,  29). 
With  i^  as  a  center,  and  FA  as 
a  radius,  describe  a  circumference. 
BGHA  is  equal  to  the  given  angle  G. 

For  AD,  being  perpendicular  to  the  radius  FA,  is  a  tangent.     Any 


Any  angle  inscribed  in  the  arc 


OF  THt 

MVERS 

Akt.  83.]    r.^^^^^^^,^x^h^IRCUMFERENCES.  65 

angle  contained  in  the  arc  AEOB  is  therefore  equal  to  BAD,  which  was 
made  equal  to  G  (22  and  23). 


34.  Exercises  in  Drawing. — Many  geometers  have  ad- 
vised the  use  of  what  is  called  the  analytic  method  for  discov- 
ering the  solutions  of  exercises  in  drawing. 

Its  steps  are  :  1.  Suppose  the  problem  solved  and  the  figure 
completed.  2.  Find,  with  or  without  the  aid  of  auxiliary  lines, 
the  geometrical  relations  of  the  parts  of  the  completed  figure  ; 
and,  3.  Make  a  solution  from  these  principles. 

However,  the  truth  and  the  proof  can  generally  he  discov- 
ered as  readily  hy  reasoning  from  what  is  given  as  hy  heginning 
with  the  figure  completed.  When  the  student  does  not  succeed 
hy  one  method,  he  may  try  the  other.  No  method  can  give 
success  without  rigorous  thought  and  a  thorough  knowledge  of 
principles. 

The  term  analytic  is  also  applied  to  a  mode  of  discovering 
solutions  of  problems  by  the  use  of  algebraic  formulas. 

A  Determinate  Problem  is  one  that  admits  of  a  definite 
number  of  solutions  ;  as  the  problem,  to  draw  a  circumference 
through  two  given  points,  with  a  given  radius.  A  problem  is 
indeterminate  when  it  admits  of  an  indefinite  number  of  solu- 
tions. 

Every  locus  is  an  answer  to  an  indeterminate  problem.  For 
instance,  to  find  a  point  at  a  given  distance  from  a  given  line  is 
an  indeterminate  problem.  There  are  an  infinite  number  of 
such  points.  Their  locus  is  the  answer.  When  two  conditions 
are  named  in  a  problem,  each  may  determine  a  locus,  and  the 
point  sought  is  the  intersection  of  the  loci.  Such  a  problem  is 
determinate.  For  instance,  Problem  VII  in  the  preceding  Arti- 
cle. 

In  such  cases,  there  may  be  more  than  one  point  of  intersec- 
tion of  the  loci,  and  there  may  be  none.  In  the  former  case,  the 
problem  has  more  than  one  solution,  in  the  latter,  it  is  impossi- 
ble. One  problem  may  present  several  cases.  The  complete 
discussion  requires  a  statement  of  all  the  possible  cases  and  the 
conditions  of  each. 


66  PLANE  GEOMETRY.  [Chap.  IV. 

Exercises. — 1.  To  draw  an  angle  equal  to  the  difference  of  two 
given  angles. 

2.  To  draw  an  angle  of  45*. 

8.  From  a  given  point,  to  draw  the  shortest  line  possible  to  a  given 
straight  line. 

4.  To  find  a  point  in  a  given  straight  line,  at  a  given  distance  from 
another  given  straight  line.     [Find  the  locus.] 

5.  To  find  a  point  in  a  given  straight  line  at  equal  distances  from  two 
other  straight  lines.     [Ill,  14,  ii.] 

6.  With  a  given  length  of  radius  to  draw  a  circumference  through 
two  given  points.     [Third  Postulate  of  Euclid.] 

7.  From  two  given  points,  to  draw  two  equal  straight  lines  ending  at 
the  same  point  of  a  given  line.     [Find  the  point  in  the  given  line.] 

8.  From  two  points  on  the  same  side  of  a  straight  line,  to  draw 
straight  lines  that  meet  in  the  first  and  make  equal  angles  with  it.  [Ill,  7, 
IX,  and  III,  10,  iv.] 

9.  From  a  given  point  out  of  a  straight  line,  to  draw  a  second  line 
making  a  required  angle  with  the  first.     [Ill,  22.] 

10.  To  draw  a  line  through  a  point  such  that  the  perpendiculars  upon 
this  line,  from  two  other  points,  are  equal.     [Join  the  given  points.] 

11.  To  draw  a  circumference  with  a  given  radius,  and 

a.  Through  a  given  point  and  tangent  to  a  given  line ;  or 
5.  Touching  two  given  lines;  or 

c.  Touching  two  given  circumferences.  [In  every  case  the  center  is 
the  intersection  of  loci.] 

12.  To  draw  a  circumference  touching  a  given  line  at  a  given  point 
(Article  30,  Exercise  3),  and 

a.  Having  a  given  radius ;  or 

5.  Passing  through  a  second  given  point. 

13.  To  draw  a  circumference  through  two  given  points,  with  the  cen- 
ter in  a  given  line. 


CHAPTER  y. 
TRIANGLES, 

Article  1. — Next  in  order  is  the  consideration  of  plane 
figures  that  inclose  a  surface  ;  and,  first,  of  those  whose  bound- 
aries are  straight  lines. 

Less  than  three  straight  lines  can  not  inclose  a  surface,  for 
two  can  have  only  one  common  point.  Therefore,  the  triangle 
is  the  simplest  polygon.  From  a  consideration  of  its  properties 
those  of  all  other  polygons  are  derived. 


An  Acute-angled  triangle  has  all  its  angles  acute. 

A  Right-angled  triangle  has  one  of  the  angles  right. 

An  Obtuse-angled  triangle  has  one  of  the  angles  obtuse. 

The  Hypotenuse  of  a  right-angled  triangle  is  the  side  that 
subtends  the  right  angle. 

An  Equilateral  triangle  has  three  sides  equal. 

An  Isosceles  triangle  has  only  two  sides  equal. 

A  Scalene  triangle  has  no  two  sides  equal. 

The  angles  at  the  ends  of  one  side  of  a  triangle  are  said  to 
be  adjacent  to  that  side.  The  angle  formed  by  the  other 
two  sides  is  opposite.  Thus,  a  side  subtends  its  opposite 
angle. 

The  Altitude  of  a  triangle  is  the  perpendicular  distance  be- 
tween one  side  and  the  vertex  of  the  opposite  angle.     This  side 


68  PLANE  GEOMETRY.  [Chap.  V. 

is  called  the  base,  and  the  opposite  vertex  is  called  the  vertex 
of  the  triangle. 

Any  side  of  a  triangle  may  be  taken  as  the  base.  Conse- 
quently the  altitude  may  be  any  one  of  three  distances. 

When  two  sides  of  a  triangle  have  been  mentioned,  as  in  the 
case  of  the  isosceles  triangle,  the  remaining  side  is  often  called 
the  base.  In  the  same  manner,  after  one  side  has  been  spoken 
of,  as  the  base  or  the  hypotenuse,  the  other  two  are  sometimes 
called  the  sides. 

"When  one  of  the  angles  at  the  base  is  obtuse,  the  perpen- 
dicular from  the  vertex  falls  out- 
side of  the  triangle. 

Corollary, — The  altitude  of 
a  triangle  is  equal  to  the  dis- 
tance between  the  base  and  a  line 
thrpugh  the  vertex  parallel  to  the 
base. 

A  Medial  of  a  triangle  is  a 
line  from  the  vertex  to  the  mid- 
point of  the  base.     Hence  there  may  be  three  medials  in  a  tri- 
angle. 

luscribed  and  Circumscribed. 

2.  When  a  circumference  passes  through  the  vertices  of  all 
the  angles  of  a  polygon,  the  circle  is  said  to  be  circumscribed 
about  the  polygon,  and  the  polygon  to  be  inscribed  in  the 
circle.  When  every  side  of  a  polygon  is  tangent  to  a  circum- 
ference, the  circle  is  inscribed  and  the  polygon  circum- 
scribed. 

A  circle  that  touches  one  of  the  three  sides  of  a  tri- 
angle and  the  other  two  sides  produced  is  called  an  escribed 
circle. 

Problem, — About  every  triangle  there  may  he  a  circumscribed 
circle. 

For  a  circumference  may  pass  through  any  three  points  not 
in  the  same  straight  line  (IV,  2). 


Art.  2.] 


TRIANGLES. 


69 


Corollary. — The  three  lines  that  bisect  perpendicularly  the 
three  sides  of  a  triangle  meet  in  one  point,  the  center  of  the 
circumscribed  circle  (lY,  11,  i). 


3,  Problem. — In  every  triangle  there  may  he  an  inscribed 
circle. 

In  the  triangle  AB  G,  let  lines  bisecting  the  angles  A  and  B 
be  produced  until  they  meet. 

The  point  i>,  where  the  two 
bisecting  lines  meet,  is  equally 
distant  from  the  sides  AB  and 
B  (7,  since  it  is  a  point  of  the  line 
which  bisects  the  angle  B  (III, 
13).      For  a  similar  reason,  the 

point  B  is  equally  distant  from  the  sides  AB  and  A  G. 
is,  it  is  equally  distant  from  the  three  sides  of  the  triangle. 

Therefore,  a  circle  having  I)  as  its  center,  with  a  radius 
equal  to  the  distance  from  B  to  either  side,  is  the  inscribed 
circle. 

Corollary. — The  three  lines  that  bisect  the  several  angles 
of  a  triangle  meet  at  one  point,  the  center  of  the  inscribed  cir- 
cle. 


That 


Sum  of  the  Angles, 

4,  Theorem. — The  sum  of  the  angles  of  a  triangle  is  equal 
to  two  right  angles. 

Let  the  line  BU  pass  through  the  vertex  of  one  angle,  B, 
parallel  to  the  opposite  side,  A  G. 

Then  the  angle  A  is  equal  to 
its  alternate  angle  BBA  (III,  22, 
ii).  For  the  same  reason,  the  an- 
gle G  is  equal  to  the  angle  EBG. 
Hence,  the  three  angles  of  the 
triangle  are  equal  to  the  three 
consecutive  angles  at  the  point  _B, 

whose  sum  is  two  right  angles  (III,  7,  ii).     Therefore,  the  sum 
of  the  three  angles  of  the  triangle  is  two  right  angles. 


70 


PLANE  GEOMETRY, 


[Chap.  V. 


Corollaries. — I.  Every  angle  of  a  triangle  is  the  supplement 
of  the  sum  of  the  other  two. 

II.  If  one  side  of  a  triangle  is  produced,  the  exterior  angle 
is  equal  to  the  sum  of  the  two  interior  angles  not  adjacent  to  it, 
and  is  greater  than  either  one  of  them.  Thus  -S  Cl>  is  equal  to 
the  sum  of  A  and  B. 

III.  In  every  triangle,  at 
least  two  of  the  angles  are 
acute. 

lY.  If  two  angles  of  a  tri- 
angle are  equal,  they  are  both 
acute. 

V.  In  a  right-angled  trian- 
gle, the  two  acute  angles  are  complementary. 

VI.  If  two  angles  of  a  triangle  are  respectively  equal  to  two 
angles  of  another,  then  the  third  angles  are  also  equal. 


Sides. 

5.  Theorem. — Each  side  of  a  triangle  is  less  than  the  sum 
of  the  other  two,  and  greater  than  their  difference. 

The  first  part  of  this  theorem  is  an  immediate  consequence 
of    the   Axiom   of    Distance ; 
that  is, 

AC<AB  +  Ba 

Subtract  AB  from  both 
members  of  this  inequality, 
and 

AG  —  AB<BO. 
That  is,  BC  is  greater  than  the  difference  of  the  other  sides. 


6.   Theorem. —  When  two  sides  of  a  triangle  are  equal,  the 
angles  opposite  to  them  are  equal ;  and  conversely. 
Circumscribe  a  circle  about  the  triangle. 


I 


Abt.  6.] 


TRIANGLES, 


n 


The  equal  sides  A  C  and  B  G 
are  chords  subtending  equal  arcs 
AEG  and  BDG  (IV,  12,  iii). 
The  angles  A  and  B,  subtended 
by  these  equal  arcs,  are  equal  (lY, 
22,  ii). 

Conversely,  if  the  angles  A 
and  B  are  equal,  their  subtend- 
ing arcs  must  be  equal,  and  have 
equal  chords. 

Corollaries. — I.  An  equilateral  triangle  is  equiangular  ;  and 
conversely. 

II.  A  line  from  the  vertex  of  an  isosceles  triangle  to  the  mid- 
point of  the  base  is  perpendicular  to  the  base  and  bisects  the 
angle  at  the  vertex  ;  for  a  line  from  G  to  the  midpoint  of  the 
arc  AB  is  a  diameter,  and  has  the  properties  stated.  A  line 
having  any  two  of  these  conditions  must  have  the  other  two. 


7.  Theorem. —  When  two  sides  of  a  triangle  are  unequal, 
the  angle  opposite  to  the  greater  side  is  greater  than  the  angle 
opposite  to  the  less  side  ;  and  conversely. 

If  BG  is  greater  than  GD, 
it  is  to  be  proved  that  the  angle 
GDB  is  greater  than  B.  Take 
GG  equal  to  GD,  and  join  DG. 

The  angles  GDG  and  GGD 
are  equal  (6),  but  GGD,  being  ex- 
terior to  the  triangle  BJ)G,  is 
greater  than  B  (4,  ii). 

Conversely,  if  the  angle  B  of 
the  triangle  is  greater  than  B,  the  side  that  subtends  B  can  not 
be  less  than  the  side  that  subtends  B,  for  then  the  angle  B 
would  be  less  than  J?,  as  just  demonstrated.  Neither  can  those 
two  sides  be  equal,  for  then  B  would  be  equal  to  B  (6).  There- 
fore, the  side  that  subtends  the  greater  angle  must  be  greater 
than  the  side  that  subtends  the  less. 

Corollary. — If  one  angle  of  a  triangle  is  either  obtuse  or 
right,  the  side  that  subtends  it  is  the  longest  side. 


73  PLANE  GEOMETRY,  [Chap.  V. 

8.  Exercises. — 1.  If  two  sides  of  a  triangle  are  produced,  the  lines 
that  bisect  the  two  exterior  angles  and  the  third  interior  angle  all  meet 
in  one  point.     [Same  reasoning  as  for  Article  3.] 

2.  How  many  circles  can  be  made  tangent  to  three  given  lines  ? 

3.  Demonstrate  Article  4  by  means  of  V,  2,  and  IV,  22.  Also  the 
first  clause  of  Article  6  by  III,  10,  v. 

4.  How  many  degrees  are  there  in  an  angle  of  an  equilateral  tri- 
angle ? 

5.  If  one  of  the  angles  at  the  base  of  an  isosceles  triangle  is  double 
the  angle  at  the  vertex,  how  many  degrees  in  each  ? 

6.  If  the  angle  at  the  vertex  of  an  isosceles  triangle  is  double  one  of 
the  angles  at  the  base,  what  is  the  angle  at  the  vertex  ? 

7.  If,  from  any  point  within  a  triangle,  lines  are  made  to  the  two  ends 
of  the  base,  the  angle  which  they  form  is  greater  than  the  angle  at  the 
vertex.     [V,  4,  ii,  or  IV,  26,  i.] 

8.  If,  from  one  end  of  the  base  of  an  isosceles  triangle,  a  line  is  made 
perpendicular  to  the  opposite  side,  the  angle  made  by  that  line  and  the 
base  is  half  of  the  angle  at  the  vertex. 

9.  Can  a  triangular  field  have  one  side  436  yards,  the  second  547 
yards,  and  the  third  984  yards  long  ? 

10.  Tiie  sum  of  the  distances  from  any  point  within  a  triangle  to  the 
three  vertices  is  less  than  the  pe- 
rimeter and  greater  than  the  semi- 
perimeter.     [Ill,  3,  and  V,  6.] 

11.  If,  with  the  vertex  of  an, 
isosceles  triangle  as  a  center,  a  cir- 
cumference is  made  cutting  the 
base,  or  the  base  produced,  then 
the  segments  of  the  base  cut  off 
by  the  curve  are  equal.  [Prove 
AB=^CD,&\aoBE=^FG.^ 


Equality  of  Triangles. 

9.  The  three  sides  and  the  three  angles  of  a  triangle  are 
called  its  six  elements. 

Theorem, —  When  two  triangles  are  equals  every  element  of 
one  is  equal  to  the  corresponding  element  of  the  other. 

This  is  a  corollary  of  Article  4,  Chapter  II. 

Corollary. — Conversely,  if  the  sides  and  the  angles  of  two 
triangles  are  respectively  equal,  the  triangles  are  equaL 


Art.  10.] 


TRIANGLES. 


73 


Three  Sides  £qual. 

10.  Theorem. — Tkoo  triangles  are  equal  when  the  three  sides 
of  the  one  are  respectively  equal  to  the  three  sides  of  the  other. 

Let  the  side  JBD  be  equal 
to  Aly  the  side  B  G  equal  to 
AJE,  and  CD  to  EI. 

Apply  the  line  AI  to  its 
equal  BD,  the  point  A  upon 
B,  Then  I  falls  upon  JD, 
since  the  lines  are  equal. 

Let  the  triangles  lie  on 
opposite  sides  of  this  common 
line,  as  in  the  second  diagram. 
Join  CK 

By  hypothesis,  the  points  C  and  E  are  equally  distant  from 
I),  and  also  equally  distant 
from  B.  Therefore,  BB  bi- 
sects perpendicularly  the  line 
CE  (III,  11,  c).  Then,  one 
of  the  given  triangles  being 
turned  upon  BD  as  an  axis, 
the  angles  at  F  being  right 
angles,  the  point  C  must  fall 
upon  jK  Thus  the  triangles 
coincide  and  are  equal. 

Two  Sides  and  the  Included  Angle. 

11,  Theorem. — Two  triangles  are  equal  when  they  have  two 
sides  and  the  included  angle  of  one  respectively  equal  to  two  sides 
and  the  included  angle  of  the  other. 


Suppose  the  angle  at  A  equal  to  the  angle  Z>,  the  side  AB 
to  the  side  BF,  and  AC  to  BE, 


74 


PLANE  GEOMETRY. 


[Chap.  V. 


Apply  the  side  AG  to  its  equal  DE.     Since  the  angles  A 
and  I)  are  equal,  AB  takes  the  direction  DF,  and,  these  lines 


being  equal,  B  falls  upon  F,     Therefore,  B  G  and  FF  coincide, 
and  the  triangles  coincide  throughout. 

One  Side  and  Two  Angles. 

13,  Theorem. — Two  triangles  are  equal  when  one  side  and 
two  angles  of  one  are  respectively  equal  to  the  corresponding  ele- 
ments of  the  other. 


Suppose  the  side  AG  equal  to  BF,  and  any  two  of  the 
angles  A,  B,  and  G  equal  respectively  to  the  corresponding 
angles  of  the  other  triangle.  Then  the  third  angles  must  be 
equal  also  (4,  yi). 

Apply  the  side  AG  to  its  equal  BF,  so  that  the  vertices  of 
the  equal  angles  come  together,  A  upon  B,  and  G  upon  F,  and 
so  that  both  triangles  fall  upon  one  side  of  the  common  line. 

Then,  since  the  angles  A  and  B  are  equal,  AB  takes  the 
direction  BF,  and  the  point  B  falls  somewhere  in  the  line  BF. 
Since  the  angles  G  and  F  are  equal,  GB  takes  the  direction 
FF,  and  B  is  also  in  the  line  FF.  Therefore,  B  falls  upon  F, 
the  only  point  common  to  the  lines  BF  and  EF  Hence,  the 
sides  of  the  one  triangle  coincide  with  those  of  the  other,  and 
the  two  triangles  are  equal. 


Art.  12.]  TRIANGLES.  75 

Corollary. — Two  right-angled  triangles  are  equal,  when  an 
acute  angle  and  any  side  of  one  are  equal  to  the  corresponding 
elements  of  the  other. 


13.  Scholium. — Two  equal  triangles  may  be  so  placed  as  to 
be  symmetrical  figures,  either  about  an  axis  or  about  a  center  of 
symmetry. 

In  the  diagram  of  Article  10  the  triangles  are  symmetrical 
with  reference  to  an  axis  of  symmetry. 

in  Article  12  the  triangles  are  symmetrical  with  reference  to 
a  point  at  the  intersection  of  the  lines  AD  and  £M 

In  Article  10  the  superposition  is  effected  by  turning  one  tri- 
angle around  the  axis  of  symmetry  ;  in  Article  11,  by  sliding 
one  triangla  along  the  plane  without  rotation  of  either  kind  ;  and 
in  Article  12,  by  turning  one  triangle  in  the  plane  and  around 
the  center  of  symmetry. 

In  some  positions  of  figures,  all  three  of  these  motions  are 
requisite  for  superposition.     Compare  III,  31. 

Two  Sides  and  an  Opposite  Angle. 

14.  Theorem. — Two  triangles  are  equal  when  one  of  them 
has  two  sides,  and  the  angle  opposite  to  the  side  which  is  not  less 
than  the  other  given  side  respectively  equal  to  the  corresponding 
elements  of  the  other  triangle. 

Let  the  sides  AE  and  EI^  EI  being  equal  to  or  greater  than 
AE,  and  the  angle  A,  be  respectively  equal  to  the  sides  B  G, 
CD,  and  the  angle  B. 


Let  the  side  AE  be  placed  on  its  equal  BG.     Since  the 
angles  A  and  B  are  equal,  AI  takes  the  direction  BD^  and  the 


76 


PLANE  GEOMETRY. 


[Chap.  V. 


point  I  falls  on  the  line  JBI>.  Since  EI  and  CD  are  equal,  the 
point  I  is  in  the  circumference  whose  center  is  at  (7,  and  whose 
radius  is  equal  to  CD.  Now,  this  circumference  cuts  a  straight 
line  extending  from  B  toward  D  in  only  one  point ;  for  B  is 


either  within  or  on  the  circumference,  since  DC  ib  equal  to  or 
less  than  CD.  Therefore,  the  point  I  falls  on  Z>,  AI  and  DD 
are  equal,  and  the  triangles  are  equal  (10). 

Corollary. — Two  triangles  are  equal  when  they  have  an 
obtuse  or  a  right  angle  and  any  two  sides  in  one  respectively 
equal  to  the  corresponding  elements  in  the  other  (7,  c). 


Exceptions  to  the  General  Hule. 


-  r~>tLk 


—     \ 


15,  Three  elements  are  always  necessary,  and  they  are 
usually  enough  to  determine  the  triangle.  There  are  two  excep- 
tions to  this  rule. 

1.  When  the  three  angles  are  given.     Two  unequal  triangles 
may  have  their  angles  respectively  equal. 


! 


2.  When  two  unequal  sides  and  the  angle  opposite  to  the  less 
are  given.  With  the  sides  AD  and  B  C  and  the  angle  A,  there 
are  two  triangles,  AB  C  and  ABD, 


Art.  16.] 


TRIANGLES. 


77 


Unequal  Triangles. 

16.  Theorem, —  When  two  triangles  have  two  sides  of  one 
respectively  equal  to  two  sides  of  the  other,  and  the  included 
angles  unequal,  the  third  side  in  that  triangle  that  has  the  greater 
angle  is  greater  than  in  the  other. 

Let  B  CD  and  AEI  be  two  triangles,  having  B  C  equal  to 
AE,  BD  equal  to  AI,  and 
the   angle  A   less   than  B, 
It  is  to  be  proved  that  CD 
is  greater  than  EI. 

Apply  the  triangle  AEI 
to  BCD,  making  AE  co- 
incide with  its  equal  BC, 
Since  the  angle  A  is  less  than 
B,  the  side  ^Z  falls  between 
B  C  and  BD,  in  the  position 
B  G,  and  EI  has  the  posi- 
tion CG, 

Join  GD,  and  join  B  with  K,  the  midpoint  of  GD,  Now, 
BG  is  equal  to  BD,  Therefore,  BE  is  perpendicular  to  GD 
(III,  11,  c).  Since  the  points  C  and  G  are  on  the  same  side  of 
BE,  CD  is  greater  than  CG  (III,  11),  or  its  equal  EI, 


17.  Theorem. —  Conversely,  if  two  triangles  have  two  sides 
of  one  equal  to  two  sides  of  the  other,  and  the  third  sides  un- 
equal, then  the  angle  subtended  by  the  greater  side  is  greater  than 
the  angle  subtended  by  the  less  side. 

For,  if  it  were  less,  the  opposite  side  would  be  less,  and,  if  it 
were  equal,  the  opposite  sides  would  be  equal ;  both  of  which 
conclusions  are  contrary  to  the  hypothesis. 


I 


18.  'Exercises. — 1.  The  lines  that  bisect  the  angles  at  the  base  of 
an  isosceles  triangle,  and  extend  to  the  other  sides  of  the  triangle,  are 
equal. 

2.  If,  from  the  midpoint  of  the  base  of  any  triangle,  lines  are  made 
parallel  to  the  other  sides,  and  if  a  line  is  made  joining  the  points  where 
these  parallels  reach  the  sides  of  the  triangle,  the  triangle  is  divided  into 
four  equal  parts. 


78  PLANE  GEOMETRY.  [Chap.  V. 

3.  The  two  taDgents  to  a  circle  from  one  point — that  is,  the  lengths 
from  the  common  point  to  the  points  of  contact — are  equal. 

4.  If  two  triangles  have  two  sides  and  an  opposite  angle  in  one  equal 
respectively  to  corresponding  elements  in  the  other  triangle,  if  the  angles 
opposite  to  the  other  two  equal  sides  are  not  equal,  they  are  supplement- 
ary.    [Consider  the  second  diagram  of  Article  15.] 

5.  If,  from  any  point  within  a  circle,  except  the  center,  lines  are  made 
to  various  points  on  the  circumference,  not  more  than  two  such  lines  can 
be  equal. 

Similarity  of  Triangles. 

19,  By  the  definition  of  similarity  (II,  3  and  14),  two 
figures  are  similar  when  every  angle  that  can  be  formed  by  join- 
ing points  of  one  has  its  corresponding  equal  angle  in  the 
other. 

It  has  been  usual  to  define  similar  triangles  as  those  whose 
angles  are  respectively  equal,  and  whose  homologous  sides  have 
the  same  ratio  ;  but  it  is  demonstrated  in  all  works  on  geometry 
that  two  triangles  that  agree  in  either  one  of  these  respects 
must  agree  also  in  the  other.  It  is  also  true,  and  will  be  demon- 
strated in  the  sequel,  that  when  any  two  figures  have  every  pos- 
sible angle  in  one,  formed  by  diagonals  or  other  lines,  equal 
to  its  corresponding  angle  in  the  other,  such  figures  have  all 
their  homologous  lines  proportional.  This  shows  that  the  ancient 
definition  is  redundant.  .^ f3^A    CL    - 

Angles  Equal. 

30.  Theorem.. —  Two  triangles  are  similar  when  two  angles 
of  one  are  respectively  equal  to  two  angles  of  the  other. 

It  follows  immediately  that  the  third  angles  are  equal  (4,  vi), 
that  is,  all  the  angles  formed  by  the  sides  of  the  triangles  are 
respectively  equal. 

It  is  to  be  shown  that  any  other  angle  in  one  of  the  trian- 
gles has  its  homologous  equal  angle  in  the  other. 

Let  the  angles  A,  B,  and  C  be  respectively  equal  to  the 
angles  JD,  E^  and  F,  and  suppose,  first,  that  the  angle  to  be  con- 
sidered has  one  arm  passing  through  a  vertex  of  the  triangle,  as 


Art.  20.]  TRIANGLES,  79 

IG.     From  the  point  F,  homologous  to  G,  make  FZ,  making 
the  angle  LFF  equal  to  IGB, 


Subtracting  these  from  the  given  equal  angles  F  and  (7,  the 
remainders  DFL  and  A  GI  are  equal.  Since  the  angles  A  and 
D  are  equal,  DLF  must  be  equal  to  AIG  (4,  yi),  and  FLE 
must  be  equal  to  GIB. 

Suppose  now  a  line  that  does  not  pass  through  one  of  the 
vertices,  as  RI.  First  connect  IG^  and  make  FL  as  before,  and 
XJf  with  the  angle  JliXii^  equal  to  RIG.  It  is  shown,  by  reason- 
ing as  above,  that  every  angle  made  at  R  or  at  Zhas  its  homolo- 
gous equal  angle  at  M  ox  at  L. 

Therefore,  the  relative  directions  of  all  their  points  are  the 
same  in  both  triangles  ;  that  is,  they  have  the  same  form,  they 
are  similar  triangles. 

Corollaries. — I.  Two  similar  triangles  may  be  divided  into 
the  same  number  of  triangles  respectively  similar  and  similarly 
arranged. 

II.  If  two  sides  of  a  triangle  are  cut  by  a  line  parallel  to  the 
third  side,  the  triangle  cut  off  is  similar  to  the  original  triangle 
(III,  22). 

This  is  true  when  the  sides  are  divided  externally  as  well  as 
when  the  division  is  internal ;  and  the  external  division  may  be 


\7 


on  the  sides  produced  either  beyond  the  base  or  beyond  the 
vertex. 


80 


PLANE  GEOMETRY. 


[Chap.  V. 


III.  Two  right-angled  triangles  are  similar  when  an  acute 
angle  in  one  is  equal  to  one  in  the  other. 

31.  Theorem. — Two  triangles  are  similar,  lohen  the  sides 
of  one  are  parallel  to  those  of  the  other ;  or,  when  the  sides  of 
one  are  perpendicular  to  those  of  the  other. 

The  angles  formed  by  lines  that  are  parallel  are  either 
equal  or  supplementary ;  and  the  same  is  true  of  angles  whose 
arms  are  perpendicular.  It  is  to  be  shown  that  the  angles  can 
not  be  supplementary  in  two  triangles. 

If  even  two  angles  of  one  triangle  could  be  respectively  sup- 
plementary to  two  angles  of  another,  the  sum  of  these  four 
angles  would  be  four  right  angles  ;  and  then  the  sum  of  all  the 


angles  of  the  two  triangles  would  be  more  than  four  right 
angles,  which  is  impossible  (4).  Hence,  when  two  triangles 
have  their  sides  respectively  parallel  or  perpendicular,  two  of 
the  angles  of  one  triangle  must  be  equal  to  two  of  the  other. 
Therefore,  the  triangles  are  similar. 

Sides  Proportional. 

3 2.   Theorem. —  When  two  triangles  are  similar,  every  side 
of  one  has  the  same  ratio  to  the  homologous  side  of  the  other. 


r    B 


Suppose  the  angles  A^  E,  and  I  respectively  equal  to  B,  C, 


Art.  22.]  TRIANGLES.  81 

and  D,  Then  AE  and  B  C  are  homologous  sides,  also  EI  and 
GB. 

Take  CF  equal  to  EA^  and  (76^  equal  to  EI^  and  join  EG. 

The  angle  Ci^6r  is  equal  to  ^  (H)*  3'^<i  therefore  to  B. 
Then  i^6?  is  parallel  to  BJD  (III,  24).  Making  a  line  parallel 
to  these  through  (7,  the  three  parallel  lines  are  cut  by  two 
secants,  GB  and  GD.  Therefore,  these  secants  are  cut  propor- 
tionally (III,  28),  and 

GFi  GB  =  GG'.  GD, 
Therefore,  EA'.GB  =  EI',  GB. 

Similarly,  AI:BI)  =  Ei:  GB. 

Corollaries. — I.  Any  homologous  lines  in  similar  triangles 
are  proportional. 

The  Xjinear  Ratio  of  two  similar  figures  is  the  ratio  be- 
tween homologous  lines. 

II.  The  perimeters  of  similar  triangles  have  the  linear  ratio 
of  the  figures  (I,  3,  vi). 

III.  If  two  sides  of  a  triangle  are  cut  by  one  or  more  lines 
parallel  to  the  base,  the  sides  are  cut  proportionally. 


23«  Theorem. — If  a  line  cuts  two  sides  of  a  triangle  in 
direct  proportion,  the  triangle  cut  off  is  similar  to  the  original 
triangle. 

Suppose  that  the  secant  EG  cuts  the  sides  of  the  triangle  so 
that 

BE'.BG  =  BG:BB. 

Now,  if  a  line  is  made 
through  E,  parallel  to  GB, 
it  must  cut  BB  in  this  same 
ratio,  and  therefore  at  the 
same   point   G.     But  there 

can  be  only  one  straight  line  through  E  and  G.  Therefore  this 
line  is  parallel  to  GB,  and  the  triangle  BEG  is  similar  to  B  GB 
(20,  II). 

Corollary. — The  secant  line  is  parallel  to  the  base. 


82 


PLANE  GEOMETRY. 


[Chap.  V. 


24.  Theorem. — Two  triangles  are  similar,  when  two  sides 
of  one  have  respectively  to  two  sides  of  the  other  the  same  ratio 
and  the  included  angles  are  equal. 

Suppose 

AE'.BC'.'.AI'.BB, 

and  let  the  angle  A  be  equal  to  B, 

Take  BF  equal  to  AE,  and  BG  equal  to  AI,  and  join  FG, 
The  triangle  BFG  is   equal  to  AEI  (11),  and  is  similar  to 


BCD  (23).     Therefore,  the  triangles  AEI  and  BCD  are  simi- 
lar. 


25.   Theorem.. — Two  triangles  are  similar  when  every  side 
of  07ie  has  the  same  ratio  to  the  corresponding  side  of  the  other. 
Suppose 

EA:CB  =  EI:CD  =  AI:  BD. 

Take  CF  equal  to  EA,  and  CG  equal  to  EI.     Then,  sub- 
stituting these  equals  in  the  first  above  proportion, 

CF:CB=CG:CD. 


Art.  25.] 


TRIANGLES. 


83 


Join  FG.     The   triangle    CFG  is  similar  to    CBD   (24), 


and, 


By  hypothesis, 


CF'.GB  =  FG:BI>. 


FA:GB  =  AI:BJ). 


Since  GF  is  equal  to  FAj  FG  is  equal  to  AT;  and  the  tri- 
angles GFG  and  AFI  are  equal  (10).  Therefore,  the  triangles 
AFI  and  B  GD  are  similar. 


36.   Theorem., — If  from  two  vertices  of  a  triangle  perpen- 
diculars fall  on  the  opposite  sides, 
a  line  joiniyig  the  feet  of  these 
perpendiculars  cuts  off  a  triangle 
which  is  similar  to  the  first  one. 

It  is  to  be  proved  that  the  tri- 
angle BED  is  similar  to  BA  G. 

The  triangles  BDA  and  BEG, 
being  right-angled  at  D  and  E, 
and  having  the  common  angle 
B,  are  similar  (20).  Therefore, 
BA'.BG=BD\BE,  and  the 
triangles  BED  and  BGA  are 
similar  (24). 

It  may  be  necessary  to  pro- 
duce one  or  both  of  the  sides. 

The  sides  of  the  triangle 
ABG  are  inversely  proportional 
to  their  segments  EB  and  DB, 


Centers  of  Similarity. 

27.  Theorem. — If  lines  are  made  from  the  several  vertices 
of  a  triangle  to  any  point  of  the  plane,  and  these  lines  are 
divided  in  the  same  ratio,  the  points  of  division  are  the  vertices 
of  a  triangle  similar  to  the  first. 


84  PLANE  GEOMETRY,  [Chap.  V. 

From  Af  B,  and  C,  make  lines  to  G.  Take  any  point  on 
GA,  as  Z>,  and  take  the  points  E  and  F,  so  that 

GD'.GA=  GE'.  GB  =  GF-.  GO-, 

then  it  is  to  be  proved  that  BEF  is  similar  to  AB  C. 

In  the  triangle  GAB,  BE  is  parallel  to  AB  (23,  c).  Like- 
wise, EF  and  BF  are  respectively  parallel  to  BG  and  AG. 
Therefore,  the  triangle  BEF  is  similar  to  AB  C, 


The  division  may  be  external.  Then  GHI  is  similar  to 
ABG  (24)  ;  and  B[I  is  parallel  to  AB  (III,  24,  i).  In  the 
same  way  the  other  sides  are  respectively  parallel,  and  the  tri- 
angle HIK  is  similar  to  AB  C, 

28.  Theorem. —  Conversely,  when  tioo  triangles  have  their 
sides  respectively  parallel,  the  three  lines  made  through  homolo- 
gous vertices  meet  at  one  point. 

If  the  triangles  ABG  and  BEF  have  their  sides  respective- 
ly parallel,  it  is  to  be  proved  that  the  lines  AB,  BE,  and  CF 
must  meet  at  one  point. 

Produce  these  lines  till  they  meet.  Suppose  AB  and  CF 
meet  at  G,  and  AB  and  BE  meet  at  N. 

Since  the  triangles  AGC  and  B GF  are  similar  (20,  ii), 

AC'.BF=AGxBG. 

Likewise,  AB :  BE  =  AJST :  BJST. 


Art.  28.]  TRIANGLES.  85 

But  by  hypothesis, 

AG:DF=AB'.DK 
Therefore,  AG.DG^  AJST :  BJST, 

That  is,  the  line  AD  is  divided  in  the  same  ratio  at  N  and 
Gy  which  is  absurd,  unless  iV  and  G  are  one  point. 


/. 


The  demonstration  is  the  same  for  the  triangle  HIK, 

A  Center  of  Similarity  is  a  point  similarly  situated  with 
reference  to  two  similar  figures,  whose  homologous  lines  are  -^ 
parallel. 

It  is  an  internal  or  external  center  of  similarity  accord- 
ing as  it  divides  internally  or  externally  the  lines  joining  homol- 
ogous points. 

The  center  of  similarity  may  be  at  any  point  of  the  plane. 
It  may  be  at  one  of  the  vertices,  as  when  a  triangle  is  cut  by  a 
line  parallel  to  the  base  ;  or  it  may  be  on  one  side.  Either  an 
internal  or  an  external  center  of  similarity  may  be  within  both 
polygons,  or  it  may  be  within  neither.  Let  the  student  illus- 
trate each  of  these  cases. 

Corolla,ry. — If  two  triangles  have  their  homologous  sides 
parallel,  a  line  joining  homologous  points  is  divided  at  the  cen- 
ter of  similarity  in  the  linear  ratio  of  the  two  figures. 


86  PLANE  GEOMETRY.  [Chap.  V. 

Right-angled  Triangles. 

29,  Every  triangle  may  be  divided  into  two  right-angled 
triangles,  by  a  perpendicular  from  the  vertex  to  the  base.  The 
investigation  of  the  properties  of  right-angled  triangles  leads 
thus  to  many  of  the  properties  of  triangles  in  general. 

Theorem. — In  a  right-angled  triangle,  if  a  perpendicular 
falls  from  the  vertex  of  the  right  angle  upon  the  hypotenuse, 
then, 

1.  I^ach  of  the  triangles  thus  formed  is  similar  to  the  origi- 
nal triangle  ; 

2.  Either  side  of  the  original  triangle  is  a  mean  propor- 
tional between  the  hypotenuse  and  the  adjacent  segment  of  the 
hypotenuse;  and, 

3.  The  perpendicular  is  a  mean  proportional  heticeen  the  two 
segments  of  the  hypotenuse. 

The  right-angled  triangles  AEO  and  AEI  have  the  acute 
angle  A   common.     Therefore, 
these  two  triangles  are  similar 
(20,  III). 

That  the  triangles  EOI  and 
*MIA  are  similar  is  proved  by 
the  same  reasoning. 

Since  the  triangles  are  simi- 
lar, the  homologous  sides  are  proportional,  and 

AI'.AE^AE\AO\ 

that  is,  the  side  AE  is  a  mean  proportional  between  the  whole 
hypotenuse  and  the  segment  A  0  which  is  adjacent  to  that 
side. 

In  like  manner,  EI  is  a  mean  proportional  between  AI  and 
OL 

Lastly,  the  triangles  AEO  and  EIO  are  similar  (20),  and 
therefore, 

A0\  0E=^  0E\  01', 

that  is,  the  perpendicular  is  a  mean  proportional  between  the 
two  segments  of  the  hypotenuse. 


Art.  29.] 


TRIANGLES. 


87 


Corollary. — A  perpendicu- 
lar from  any  point  of  a  circum- 
ference upon  a  diameter  is  a 
mean  proportional  between  the 
two  segments  of  the  diameter 
(lY,  22,  III). 


1/  \> 


30.  Theorem. —  The  second  power  of  the  length  of  the  hy- 
potenuse is  equal  to  the  sum  of  the  second  powers  of  the  lengths 
of  the  other  two  sides  of  a  right-angled  triangle. 

Let  h  be  the  hypotenuse,  a  the  perpendicular  let  fall  upon 
it,  h  and  c  the  other  sides,  and  d  and  e  the  corresponding  seg- 


ments of  the  hypotenuse  made  by  the  perpendicular.  That 
is,  these  letters  represent  the  ratios  of  these  lines  to  some  unit 
of  length. 

By  the  second  conclusion  of  the  last  theorem, 

h'.h  '.:J)  :  d,      and       hic.-.  c.e. 
Hence,  hd=^J)^y         and  he=.c^. 

Adding  these  two,  h  (d  -\-  e)  =i  b^  •\-  c^ . 

But        c?  +  e  =  A  ;  therefore,  A2  =  ^»2  +  c^. 


31.  Theorem. — In  any  triangle^  if  a  perpendicular  falls 
from  the  vertex  upon  the  hase^  the  sum  of  the  segments  of  the 
base  is  to  the  sum  of  the  other  two  sides  as  the  difference  of 
those  sides  is  to  the  difference  of 
the  segments  of  the  base. 

Let  a  be  the  perpendicular, 
b  the  base,  c  and  d  the  sides, 
and  e  and  i  the  segments  of  the 


PLANE  GEOMETRY. 


[Chap.  V. 


Then,   two    right-angled   triangles   are   formed,   in   one   of 

which 

a2  +  ^2  =  d^  ; 


and  in  the  other, 

Subtracting, 

Factoring, 

Whence, 


a2  + 


^2— e2 

(*  +  «)  (*  — 6) 
i-\-e:d-{-c 


{d-^c)  (d  —  c). 
d — c  :  i  — e. 


The  base  is  equal  to  the  first 
or  to  the  last  term  of  this  pro- 
portion, according  as  the  perpen- 
dicular divides  it  internally  or 
externally. 

Corollary. — The     segments 

of  the  base  may  be  expressed 

in  terms   of  the   sides   of  the   triangle 

d^  —  c^ 
2  =F  e  =  — , .     Therefore, 


For   i  ±e  =  h,   and 


I  = 


53 


+  C?2 


26 


,  and  e 


h^-\-c' 


d^ 


2b 


Segments  of  Chords. 

32.  Theorem. — If  two  chords  of  a  circle  cut  each  other y  the 
segments  of  one  are  the  extremes^  and  the  segments  of  the  other 
the  means  of  a  proportion. 

If  from  a  point  without  a  circle^  two  secants  extend  to  the 
farther  side,  then  the  ichole  of  one  secaiit  and  its  exterior  part 
are  the  extremes,  and  the  whole  of  the  other  secatit  and  its  exte- 
rior part  are  the  means  of  a  proportion. 


i 


Art.  82.]  TRIANGLES.  89 

The  second  proposition  is  usually  stated  as  a  separate  theo- 
rem. It  is  that  case  of  the  first  when  the  chords  divide  each 
other  externally. 

Join  AB  and  BG.  The  triangles  AED  and  CEB  have  the 
angles  B  and  B  equal  (IV,  22),  and  the  angle  E  vertical  in  one 
case  and  common  in  the  other.  Therefore,  the  triangles  are 
similar  (20),  and 

EA:EG  =  EB:EB, 

33.  Theorem. — If  from  the  same  point  there  are  a  tangent 
and  a  secant,  the  tangent  is  a  mean  proportional  between  the 
secant  and  its  exterior  part. 

This  may  be  demonstrated  by  the  student.  For  construc- 
tion, make  chords  from  the  point  of  tangency  to  the  points 
where  the  secant  cuts  the  curve. 

Medials  of  Triangles. 

34.  Theorem. — Two  medials  of  a  triangle  cut  each  other 
in  the  same  ratio,  the  smaller  segment  being  half  of  the  larger. 

It  is  to  be  proved  that  the  medials  BE  and  CB,  of  the  trian- 
gle ABC,  are  divided  at  (9,  so  that  BO  is  half  oi  OC  and  EO 
is  half  of  OB.     Join  BE. 

Since  B  and   E  are  the 
midpoints   of  AB  and  AG,  /j 

the  triangle  ABE  is  similar  /I 

to  AB  G  (23),  and  BE  is  half  ^  /   L 

oi  BG  (22).     The  triangles  /C''l 

OEB  and  OB  G  are  mutually  /■'''''' \  / 

equiangular  (III,  22,  ii)  and  /'  \l 

similar.     Therefore, 

B0\  OG  =  EO:  0B  =  BE'.BG=1  :  2. 

35.  Theorem. — The  three  medials  of  a  triangle  cut  each 
other  in  one  point. 

For  one  is  divided  by  both  the  others  in  the  same  ratio,  and 
therefore  at  the  same  point. 
5 


90 


PLANE  GEOMETRY. 


[Chap.  V. 


JVf 


36.  Theorem. — The  medial  to  the  base  of  a  triangle  is  less 
than  half  the  sum  of  the  other  two  sides. 

Join  the  midpoint  of 
the  base,  M^  to  X,  the  mid- 
point of  the  side  £  C. 

The  triangle  MLC  is 
similar  to  AB  (7,  and  LM 
is  half  of  AB.  Therefore, 
the  sum  of  BL  and  LM'vs, 
half  of  the  two  sides,  but  BM  is  shorter  than  BLM. 

CoroUai^. — The  sum  of  the  medials  is  less  than  the  perim- 
eter of  a  triangle. 

37.  Theorem. — The  sum  of  the  medials  is  greater  than 
three  fourths  of  the  perimeter  of  a  triangle. 

The  sum  of  the  three 
lines  from  any  point  in  a 
triangle  to  the  vertices  is 
greater  than  half  of  the 
perimeter.  For  A0-{- 
OB  >AB;  OB^  OG 
>  BG;  and  0G+ AG 
"^  AG,     By  addition, 

2  (AO-i-  0B-\-  OG)>AB  +  BG+AG. 

Since  each  of  the  lines  from  0  is  two  thirds  of  a  medial,  two 
thirds  the  sum  of  the  medials  is  greater  than  half  the  perimeter. 
Therefore,  the  sum  of  the  medials  is  greater  than  three  fourths 
of  the  perimeter. 


38.  Theorem. — The  perimeter  and  three  fourths  of  the 
perimeter  are  the  limits  of  the  sum  of  the  medials  of  a  tri- 
angle. 

It  only  remains  to  be  proved  that  the  sum  of  the  medials 
may  be  within  less  than  any  assignable  difference  of  these  limits. 
The  student  may  demonstrate  the  superior  limit  by  a  triangle 
having  one  very  small  side,  and  the  inferior  limit  by  a  triangle 
with  one  very  small  medial. 


Aet.  39.] 


TRIANGLES, 


91 


39.  Theorem. —  One  medial  of  a  triangle  is  less  than  the 
sum  and  greater  than  the  difference  of  the  other  two. 

In  any  triangle  ABC^  having  the  medials  AL,  JBM,  and 
CJ^y  the  line  NL  may  be  made,  and  it  may  be  produced  to  P, 
making  LP  equal  to  iVX,  and  the  lines  BP  and  MP  may  be 
made. 

As  AM  is  half  of  A  (7,  it  is  equal  to  iVZ,  and  therefore  to 
LP,  to  which  it  is  also  parallel,  making  the  figure  AMPL  a 
parallelogram.     Therefore,  MP  is  equal  to  AL.     As  the  trian- 
gles LLP  and  CZA^have 
two    sides    and    the    in- 
cluded angle  of  one  equal 
to  the  corresponding  ele- 
ments of   the  other,   the 
third  side,  BP,  is  equal  to 
NC.     Thus,  the  triangle 
BMP  has  for  its    sides 
the  medials  of  the  given 
triangle  AB  G. 

Since    this    construc- 
tion is  possible   for  any 

given  triangle,  every  medial  must  be  less  than  the  sum  and 
greater  than  the  difference  of  the  other  two. 


40.  Applications. — Principles  demonstrated  in  this  chapter 
are  used  to  measure  heights  and  distances,  even  the  distances  of 
the  stars. 

One  side  and  the  angles  of  a  triangle  determine  the  other  sides.  By 
measuring  a  base  and  the  angles  at  each  end  made  by  lines  extending  to  a 
distant  object,  the  distance  is  ascertained.  Trigonometry  teaches  how- 
to  calculate  these  distances.  Such  problems  may  be  solved  approxi- 
mately by  drawing  a  triangle  similar  to  the  one  measured,  measur- 
ing its  sides,  and  applying  the  principle  that  homologous  lines  are 
proportional. 

Similarly,  the  height  of  any  object  may  be  found  when  its  angular 
elevation  and  its  distance  are  known.  "Without  any  drawing,  or  consider- 
ation of  the  angles,  the  height  of  a  house  or  tree  may  be  found  by  meas- 
uring its  shadow,  and  at  the  same  time  the  length  of  shadow  of  a  yard- 
stick held  vertically. 


92  PLANE  GEOMETRY.  [Chap.  V. 

41.  Exercises. — 1.  Given  tlie  sides  of  a  triangle,  17  and  18,  and 
the  base,  23 ;  to  find  the  segments  of  the  base  made  by  a  perpendicular 
from  the  vertex. 

2.  The  parts  of  two  parallel 
lines,  intercepted  by  several 
straight  lines  that  meet  at  one 
point,  are  proportional. 

3.  Find  the  locus  of  the  mid- 
points of  all  lines  that  extend 
from  a  given  point  to  a  given 
line. 

4.  If  tangents  to  two  inter- 
secting circles  are  made  from 

any  point  on  the   common  chord  produced,  such  tangents  are  equal. 
[33.] 

5.  The  common  chords  of  every  pair  of  three  intersecting  circles 
meet  in  one  point.     [32.J 


43.  Scholium. — The  cases  of  equality  of  triangles  are  : 
1.  Three  sides  equal ;  2.  Two  sides  and  included  angle ; 
3.  One  side  and  two  angles  ;  and  4.  Two  sides  and  an  oppo- 
site angle. 

The  cases  of  similarity  are  :  1.  Three  sides  proportional ;  2. 
Two  sides  proportional  and  included  angles  equal ;  3.  Two 
angles  equal  (including  the  cases  of  parallel  and  perpendicular 
sides).  A  fourth  case  might  be  stated,  when  two  sides  are  pro- 
portional, and  the  angles  opposite  the  greater  of  the  given  sides 
are  equal.  Thus,  each  case  in  the  theory  of  equality  has  its  cor- 
responding case  in  the  theory  of  similarity. 

In  every  case  of  equality  of  triangles,  at  least  one  line  in 
one  must  be  equal  to  the  corresponding  line  in  the  other.  In  no 
case  of  similarity  of  triangles  is  any  equal  dimension  given. 
Any  case  of  similarity  becomes  a  case  of  equality,  if  we  add  to 
the  hypothesis  that  some  two  homologous  lines  are  equal.  That 
is,  when  triangles  are  similar,  and  when  the  linear  ratio  is  unity, 
they  are  equal. 

In  similarity,  equality  of  corresponding  angles  is  assumed  in 
the  definition,  equality  of  homologous  ratios  is  proved  as  the 
universal  consequence. 


Art.  43.] 


TRIANGLES. 


93 


43,  Miscellaneous  Exercises. — 1.  If  the  diameter  of  a  circle 
is  one  of  tbe  equal  sides  of  an  isosceles  triangle,  the  circumference  bisects 
the  base  of  the  triangle.     [IV,  26,  ii.] 

2.  If  two  circles  have  for  their  diameters  two  sides  of  a  triangle,  the 
circumferences  cut  the  third  side  at  the  same  point.  It  may  be  necessary 
to  produce  the  third  side. 

3.  If  three  circles  have  the  vertices  of  a  triangle  as  centers,  and 
the  circumferences  pass  through  the  point  of  contact  of  the  sides 
with  the  inscribed  circle,  those  three  circles  are  tangent  each  to  the 
others. 

4.  If  an  isosceles  and  an  equilateral 
triangle  are  on  the  same  base,  and  if  the 
vertex  of  the  inner  triangle  is  equally 
distant  from  the  vertex  of  the  outer 
one  and  from  the  ends  of  the  base,  then, 
according  as  the  isosceles  triangle  is 
the  inner  or  the  outer  one,  its  base 
angle  is  i  of,  or  2^  times  its  vertical 
angle. 

6.  The  semi-perimeter  of  a  triangle  is 
greater  than  any  one  of  the  sides,  and  less 
than  the  sum  of  any  two.     [5.] 

6.  The  angle  at  the  base  of  an  isos- 
celes triangle,  being  one  fourth  of  the 
angle  at  the  vertex,  if  a  perpendicular 
is  erected  to  the  base  at  its  extreme 
point,  and  this  perpendicular  meets  the 
opposite  side  of  the  triangle  produced, 
then  the  part  produced,  the  remaining 
side,  and  the  perpendicular  form  an  equi- 
lateral triangle. 

7.  Of  all  triangles  on  the  same  base,  and  having  their  vertices  in  the 
same  line  parallel  to  the  base,  the  isosceles  has  the  greatest  vertical  angle. 
[Circumscribe  a  circle  about  the  isosceles  triangle.] 

8.  If,  from  a  point  without  a  circle,  two  tangents  are  made  to  the 
circle,  and  if  a  third  tangent  is  made  at  any  point  of  the  circumference 
between  the  first  two,  then,  at  whatever  point  the  last  tangent  is  made, 
the  perimeter  of  the  triangle  formed  by  these  tangents  is  twice  the  length 
of  one  of  the  tangents  first  made. 

9.  The  midpoint  of  the  hypotenuse  is  equally  distant  from  the  three 
vertices  of  a  right-angled  triangle. 


94  PLANE  GEOMETRY.  [Chap.  V. 

10.  If  a  circle  is  inscribed  in  a  right-angled  triangle,  the  differ- 
ence between  the  hypotenuse  and  the  sum  of  the  two  sides  is 
equal  to  the  diameter  of  the  circle.  [Make  a  radius  to  each  point 
of  tangency.] 

11.  In  a  right-angled  triangle,  if  one  of  the  acute  angles  is  equal 
to  twice  the  other,  the  hypotenuse  is  equal  to  twice  the  shortest 
Bide.  [Join  the  vertex  of  the  right  angle  to  the  midpoint  of  the 
hypotenuse.] 

12.  Two  triangles  are  similar,  when  two  sides  of  one  are  proportional 
to  two  sides  of  the  other,  and  the  angle  opposite  to  that  side  which  is 
equal  to  or  greater  than  the  other  given  side  in  one  is  equal  to  the  cor- 
responding angle  in  the  other.     [14.] 

13.  If  perpendiculars  fall  from  the  three  vertices  on  the  opposite 
sides  of  a  triangle,  the  triangle  formed  by  joining  the  feet  of  these 
perpendiculars  has  its  angles  (either  interior  or  exterior)  bisected  by 
the  perpendiculars.  [The  bisection  of  an  exterior  angle  occurs  when 
the  given  triangle  is  obtuse-angled,  as 

in  the  second  diagram  of  Article  26.] 

14.  If  an  equilateral  triangle  is  in- 
scribed in  a  circle,  and  from  any  point 
on  the  circumference  lines  extend  to 
the  three  vertices,  one  of  these  is  equal 
to  the  sum  of  the  other  two.  [lY,  22, 
n;  V,  22.] 

15.  Find  the  locus  of  the  points  such 
that  the  sum  of  the  distances  of  each 
from  the  two  arms  of  a  given  angle  is 
equal  to  a  given  line.  [The  point  on 
one  arm  of  the  angle  at  a  distance  from 

the  other  arm  equal  to  the  given  line  must  be  one  point  of  the  locus ;  the 
corresponding  point  on  the  other  arm  must  be  a  second  point.] 

16.  Find  the  locu^  of  the  points  such  that  the  difference  of  the  dis- 
tances of  each  from  two  arms  of  a  given  angle  is  equal  to  a  given 
line. 

Discuss  both  of  these  upon  the  supposition  that  two  indefinite  lines 
are  given  instead  of  two  arms  of  an  angle. 


Akt.  44.] 


TRIANGLES, 


95 


Problems  in  Drawing. 


I 


44:.  Every  case  of  equality  of  triangles  has  its  corresponding  problem 
in  drawing. 

I. —  To  draw  a  triangle  when  the  three  sides  are  given. 
Let  a,  &,  and  e  be  the  given  lines. 
Draw  the  line  IE  eqnal  to  c.  With 
7  as  a  center,  and  with  the  line  &  as 
a  radius,  describe  an  arc,  and  with  £J 
as  a  center  and  the  line  a  as  a  ra- 
dius describe  a  second  arc,  so  that 
the  two  may  cut  each  other.  Join 
Oj  the  point  of  intersection  of  these 
arcs,  with  /  and  with  K  JOE  is  the 
required  triangle. 

In  the  same  way  draw  a  triangle 
equal  to  a  given  triangle. 

The  demonstration  of  this  and  of  some  of  the  following  problems  is 
left  to  the  student. 

II. — 7o  draw  a  triangle^  two  sides  ^ 

and  the  included  angle  leing  given.  -^- 

Let  a  and  &  be  the  given  lines, 
and  E  the  angle. 

Draw  FG  eqnal  to  I.  At  C 
make  an  angle  equal  to  E.  Take 
DC  equal  to  a,  and  join  FD.  Then 
FDG  is  a  triangle  having  the  given 
elements. 

III. — To  draw  a  triangle  when  one  side  and  two  angles  a/re  given. 

If  one  of  the  angles  is  opposite  the  given  side,  find  the  supplement 
of  the  sum  of  the  given  angles;   this 

is   the   other   adjacent   angle.      Then,  « 

let  a  be  the  given  side,  and  D  and  E 
the  adjacent  angles. 

Draw  EG  equal  to  a.  At  B 
make  an  angle  equal  to  i),  and  at  G 
an  angle  equal  to  E.  Produce  the 
sides  till  they  meet  at  the  point  F. 
FBG  is  a  triangle  having  the  given 
side  and  angles. 


96  PLANE  GEOMETRY.  [Chap.  V. 

IV. — To  draw  a  triangle  when  two  aides  and  an  angle  opposite  to  one 
of  them  are  given. 

Construct  an  angle  equal  to  the  given  angle.  Lay  off  on  one  side  of 
the  angle  the  length  of  the  given  adjacent  side.  With  the  extremity  of 
this  side  as  a  center,  and  with  a  radius  equal  to  the  side  opposite  the 
given  angle,  draw  an  arc.  If  this  arc  cuts  or  touches  the  other  side  of 
the  angle,  join  the  point  of  intersection  or  tangeucy  with  that  point  which 
was  taken  as  a  center.     A  triangle  thus  formed  has  the  given  elements. 

The  student  can  better  discuss  this  problem  after  drawing  several 
triangles.  Let  the  given  angle  vary  from  y^tj  obtuse  to  very  acute;  and 
let  the  opposite  side  vary  from  being  much  larger  to  much  smaller  than 
the  side  adjacent  to  the  given  angle. 

y. — To  find  a  fourth  proportional  to  three  given  lines. 


Let  a  be  the  given  extreme,  and  5  and  e  the  given  means. 

On  a  line  extending  from  the  point  I)  take  DG  equal  to  a,  and  Bff 
equal  to  c;  from  G  draw  GF  equal  to  5;  join  DF;  from  ff  draw  J2X 
parallel  to  GF;  and  produce  ffK  and  BF  (if  necessary)  till  they  meet. 
HK  is  the  required  fourth  proportional. 

For  the  triangles  DGF  and  DRK  are  similar.     Hence, 

DG:GF=DE:HK. 
That  is,  a :  J  =  c :  HK. 

yi. — To  divide  a  line  in  given  ratios. 

Let  LD  be  the  line  to  be  divided  into  parts  proportional  to  the  lines 
a,  5,  and  c. 


I 


Akt.  44.] 


TRIANGLES. 


97 


a 


From  L  draw  the  line  LE^  making  LF  equal  to  a,  FG  equal  to  &,  and 
GE  equal  to  c.  Join  DE^  and  draw  (?/  and  FH  parallel  to  DE.  LR^ 
HI^  and  ID  are  the  parts  required  (III,  28). 

yil.—To  divide  a  line  hy  a  given  nwrriber. 

This  may  be  done  by  the  last  problem ;  bat,  when  the  line  is  small, 
the  following  method  is  preferable. 

To  divide  the  line  AB  by 
ten,  draw  A  G  indefinitely,  mak- 
ing of  it  ten  equal  parts.  Join 
BC^  and  from  the  several  points 
of  division  of  ^  C  draw  lines  par- 
allel to  AB^  and  produce  them 
to  BG.  The  parallel  nearest  to 
G  is  one  tenth  of  AB,  the  next  is  two  tenths,  and  so  on. 

This  depends  upon  similarity  of  triangles. 

This  is  the  method  employed  in  the  common  scale  used  for  draughting. 

VIII. — To  divide  a  line  externally  in  a  given  ratio. 

Let  LD  be  the  line  to  be  di- 
vided in  the  ratio  a :  &.  Draw 
LG  equal  to  5,  the  greater,  and 
from  G  take  FG,  toward  Z, 
equal  to  a.  Join  FD,  and  draw 
GE  parallel  to  it.  Then  LD  is 
divided  at  H,  so  that 

EDxEL^a-.l). 

IX. — Tojind  a  mean  'proportional  to  two  given  lines. 

Make  a  straight  line  equal  to  the  sum  of  the  two.  Upon  this  as  a 
diameter,  describe  a  semi- circumference.  Upon  this  diameter  erect  a  per- 
pendicular at  the  point  of  meeting  of  the  two  given  lines.  Produce  this 
to  the  circumference.  The  line  last  drawn  is  the  required  mean  propor- 
tional. 

X. —  To  divide  a  line  in  extreme  and  mean  ratio. 

Let  AG  \)Q  the  line.    At  G  erect  a  perpendicular,  67,  equal  to  half 
of  AG.     Join  AL     Take  ID 
equal  to  CI,  and  AB  equal  to 
AD.    The  line  AG  \&  divided  at 

the    point  B   in    extreme    and  X,- 

mean  ratio.    That  is, 

AG:AB=zAB:BG. 


as 


With  /  as  a  center  and  IG 
a   radius,    describe    an    arc 


98 


PLANE  GEOMETRY. 


[Chap.  V. 


DGE^  and  produce  AI  till  it  meets  this  arc  at  E.    Then,  -4(7  is  a 
tangent  to  this  arc,  and  therefore  (33), 


Hence, 


AE'.AG=AG'.AD. 
AG:AE—AG=AD'.AG—AD. 


But  AG  —  DE,    Therefore,  AE  —  AG  =  AB  =  AB,   and  ^C 
AD  =  BG.    Substituting  these  equals, 

AG'.AB  =  AB:BG. 


Corollary. — AD  is  divided 
externally  in  extreme  and  mean 
ratio,  at  the  point  JS',  for 

AD:  DE=  DE'.AE. 

The  numerical  value  of  these 
ratios  is  shown  as  follows.  If 
the  line  J.C  is  1,  GI  is  ^. 


Then  AI=  \/ {AGY  +  {Giy  =  y^Ii  =  i  |/5; 

BG  =  AG—AB=^--\\r5^ 


and 


XI. — To  draw  an  isosceles  triangle  having  an  angle  at  the  l>ase  twice  as 
great  as  the  angle  at  the  vertex. 

Divide  the  line  AB,  in  extreme  and  mean  ratio,  at  D.  Then  draw  the 
triangle  AGD  with  the  base 
equal  to  the  smaller  segment 
AD,  and  with  the  sides  GD 
and  GA  each  equal  to  the  larger 
segment  BD.  y'' 

Join  BG.    Since  AB\AG=  /' 

AG:  AD,   the    triangles   ABG  /' 

and  A.GD  are  similar  (24),  and  ^ j)  j^ 

the  angle  B  equals  Z)C^.     Since 

BD    equals   GD,  the  angles  B 

and  BGD  are  equal.     Therefore,  BGA  is  double  the  angle  B,  and  the 

angle  A  is  double  the  angle  DGA. 


Art.  46.]  TRIANGLES.  99 

45.  Exercises  in  Drawing. — The  suggestions  of  Article 
34,  Chapter  IV,  should  be  remembered  in  connection  with  these 
exercises. 

1.  The  base  of  an  isosceles  triangle  is  to  one  side  as  three  to  two. 
Find,  by  construction  and  measuring,  whether  the  angle  at  the  vertex  is 
acute  or  obtuse. 

2.  Construct  a  right-angled  triangle  when 

a.  An  acute  angle  and  one  side  are  given ; 

h.  An  acute  angle  and  the  hypotenuse  are  given ; 

c.  A  side  and  the  hypotenuse  are  given ;  and  when, 

d.  The  two  sides  are  given. 

3.  To  describe  a  circle  touching  three  given  straight  lines.  [In  what 
loci  must  the  center  be  ?] 

4.  To  draw  a  hne  DE  parallel  to  the  base,  BG^  of  a  triangle,  ABG^ 
making  DE  equal  to  the  sum  of  BD  and  CE.    [6.] 


A* 

//* 

-/ 

\ 

n^ 

7 

B^^^' 

o 

a 

C 

6.  To  draw  a  line  BE  parallel  to  the  base,  BG,  of  a  triangle,  ABG^ 
making  DE  equal  to  the  difference  of  BD  and  GE. 

6.  Through  a  given  point  to  draw  a  line  such  that  the  parts  of  it 
between  the  given  point  and  perpendiculars  let  fall  on  it  from  two  other 
given  points  are  equal.     [Ill,  27.] 

7.  To  describe  a  circumference  through  two  given  points  and  touch- 
ing a  given  straight  line.     [33.] 

8.  To  draw  an  isosceles  triangle  when  one  side  and  one  angle  are 
given.     [This  includes  several  cases.] 

9.  To  draw  a  triangle  with  a  given  base,  altitude,  and  angle  at  the 
vertex.     [Use  loci.  IV,  26,  iii.] 

10.  To  draw  a  triangle  with  a  given  base,  altitude,  and  radius  of  cir- 
cumscribed circle. 

11.  In  a  given  circle  to  inscribe  a  triangle  similar  to  a  given  triangle. 
[What  arc  must  subtend  each  angle  of  the  required  triangle  ?] 


100 


PLANE  GEOMETRY. 


[Chap.  V. 


JD 


12.  To  divide  a  right  angle  into  five  equal  parts.     [38,  xi.] 

13.  Througli  a  given  point  be- 
tween two  given  lines,  to  draw  a  line 
such  that  the  part  intercepted  by  the 
given  lines  is  bisected  at  the  given 
point.  [That  is,  given  the  point  P 
and  the  lines  A  G  and  BD^  to  draw  AB 
80  that  AP  =  PB. 

14.  From  a  point  without  two  lines, 

to  draw  a  line  such  that  the  part  intercepted  between  the  lines  is  equal  to 
the  part  between  the  given  point  and  the  nearest  line. 

15.  To  draw  a  triangle,  given  the  base,  one  angle  at  the  base,  and  the 
difference  of  the  sides.  [Draw  the  base  AB  and  the  angle  at  A  as  given. 
On  the  other  arm  of  the  angle  A^  take  AD  equal  to  the  given  difference. 
It  may  be  necessary  to  produce  the  arm  lelow  the  base  a  distance  equal 
to  the  given  difference.] 

16.  To  describe  a  circumference  tangent  to  three  given  equal  circum- 
ferences that  are  tangent  to  each  other. 


CHAPTER  VI. 
Q  UADRILA  TERALS. 

Article  1. — Quadrilaterals  are  classified  according  to  the 
parallelism  of  sides  and  the  equality  of  angles. 

A  Trapezoid  is  a  quadri- 
lateral that  has  two  sides  paral- 
lel. The  parallel  sides  are 
called  bases. 

No  two  of  the  angles  are 
necessarily  equal,  but  the  angles  at  the  ends  of  one  base  may  be 
equal.  Then  the  other  two  angles,  being  respectively  supple- 
ments of  these  (III,  22,  iv),  are  also  equal.  Such  a  figure  is 
symmetrical,  the  axis  of  symmetry  being  the  line  joining  the 
midpoints  of  the  bases. 

A  Parallelogram  is  a 
quadrilateral  that  has  its  oppo- 
site sides  parallel. 

Corollary. — Two    consecu- 
tive angles  of  a  parallelogram  are  supplementary,  and  the  oppo- 
site angles  are  equal  (III,  7,  vi).     Hence,  if  one  angle  is  right, 
they  are  all  right. 

A  Rectangle  is  a  right-angled  parallelogram. 


A  Rhombus,  or  Lozenge,  is  a  parallelogram  that  has  all 
its  sides  equal.  It  will  be  shown  (3)  that  this  is  a  consequence 
when  two  adjacent  sides  are  equal. 


102  PLANE  GEOMETRY.  [Chap.  VI. 

A  Square  is  a  quadrilateral  having  its  sides  equal  and  its 
angles  right.  It  will  appear  that  such  a  figure  is  a  rectangle 
with  equal  sides. 

The  Altitude  of  any  quadrilateral  having  parallel  sides  is 
the  distance  between  the  parallels.  Then  either  of  these  is 
called  the  base,  and  sometimes  the  two  are  called  the  bases. 
The  altitude  of  a  parallelogram  may  be  either  of  two  dis- 
tances. 

A  Diagonal  of  a  polygon  is  a  straight  line  joining  two 
vertices,  except,  of  course,  two  consecutive  vertices,  which  are 
joined  by  a  side. 

2.  Problem, — There  may  he  a  quadrilateral  inscribed  in  a 
circle. 

For  any  four  points  of  a  circumference  may  be  joined  by 
chords. 

Corollaries. — I.  The  opposite  angles  of  an  inscribed  quadri- 
lateral are  supplementary  (IV,  22,  iv). 

II.  When  a  quadrilateral  has  its  opposite  angles  supplement- 
ary, a  circle  may  be  circumscribed  about  it  (IV,  26,  rv). 

III.  If  the  opposite  angles  of  a  quadrilateral  are  not  supple- 
mentary, a  circle  can  not  be  circumscribed. 

IV.  The  rectangle  and  the  square  are  the  only  parallelo- 
grams that  can  be  inscribed  in  a  circle. 

Parallelograms. 

3«  Theorem.. — The  opposite  sides  of  a  poLraUelogram  are 
equal. 

For,  joining  -4C  by  a  diagonal, 
the  triangles  formed  have  the  side 
A  C  common ;  the  angles  A  GB  and 
DA  C  equal,  for  they  are  alternate  ; 
and  A  CD  and  BA  C  equal,  for  the 
same  reason.  Therefore  (V,  12),  the 
triangles  are  equal,  the  side  AD  is 
equal  to  B  (7,  and  AB  to  CD, 


AST.  8.]  QUADRILATERALS,  103 

Corollaries. — I.  When  two 
systems  of  parallels  cross  each 
other,  the  parts  of  one  system  in- 
cluded between  two  lines  of  the 
other  are  equal. 

II.  A  diagonal  divides  a  parallelogram  into  two  equal  tri- 
angles. 


4,  Theorem, — If  the  opposite  sides  of  a  quadrilateral  are 
equal,  the  figure  is  a  parallelogram. 

Join  A  C,     The  triangles  AJS  G  and  CD  A  are  equal,  for  the 
side  AD  is  equal  to  J3  (7,  and  D  O 
is   equal  to  AJB,  by  hypothesis  ; 

and  they  have  the  side  A  C  com-  /    ^ 

mon.    Therefore,  the  angles  DA  O        ^ ^^ 

and  BOA  are  equal.     But  these 
angles  are  alternate  with  reference 

to  the  lines  AD  and  B  G,  and  the  secant  A  G.  Hence,  AD  and 
j5  G  are  parallel,  and,  for  a  similar  reason,  AD  and  D  G  are  par- 
allel ;  that  is,  the  figure  is  a  parallelogram. 


5,  Theorem. — If  two  sides  of  a  quadrilateral  are  equal  and 
parallel,  the  figure  is  a  parallelogram. 

If  AD  and  DG  are  equal  and  parallel,  it  is  to  be  proved 
that  AB  is  parallel  to  D  G. 

Joining    BD,     the     triangles 

formed  are  equal,  since  they  have  v -■-""' \ 

the  side   BD  common,  the   side  \  ----"T"" A 

AD  equal  to  BG,  and  the  angle  ^  ^ 

ADB  equal  to  its  alternate  DB  G 

(V,  11).  Hence,  the  angle  ABD  is  equal  to  BDG.  But  these 
are  alternate  with  reference  to  the  lines  AB  and  D  G,  and  the 
secant  BD. 

Therefore,  AB  and  D  G  are  parallel,  and  the  figure  is  a  par- 
allelogram. 


104 


PLANE  GEOMETRY. 


[Chap.  VI. 


6,  Theorem. — The  diagonals  of  a  parallelogram  bisect  each 
other,  and  conversely. 

For  the  triangles  ABH 
and  CDH  have  the  sides 
AB  and  CD  equal,  the  an- 
gles ABU  and  CDH  equal, 
and  the   angles   BAH  and 

DGII  equal.     Therefore,  the  triangles  are  equal  and  AH  is 
equal  to  CH,  and  BH  to  DH. 

The  converse  may  be  demonstrated  by  the  student. 

Corollary. — The  midpoint  of  the  diagonals  of  a  parallelo- 
gram is  a  center  of  symmetry  for  the  figure  (IV,  5). 


7.  Applications. — The  rectangle  is  the  most  frequently  used  of  all 
quadrilaterals.  "Walls  and  floors  of  apartments,  doors  and  windows, 
books,  paper,  and  many  other  articles,  have  this  form. 

Carpenters  make  an  ingenious  use  of  a  geometrical  principle  in  order 
to  make  door  and  window  frames  rectangular.     Having  made  the  frame, 
with  its  sides  equal  and  its  ends  equal,  they  measure  the  two  diagonals, 
and  make  the  frame  take  such  a 
shape    that    these    also    become 
equal. 

A  rhombus  inscribed  in  a 
rectangle  is  the  basis  of  many  or- 
naments used  in  architecture  and  other  work. 

The  symmetrical  trapezoid  is  used  in  architecture,  sometimes  for  or- 
nament, and  sometimes  as  the  form  of  the  stones  of  an  arch. 


A 


B 


An  instrument  called  parallel  rulers^  used  in  drawing  parallel  lines, 
consists  of  two  rulers,  connected  by  cross  pieces  with  pins,  on  which  the 
rulers  may  turn.    The  distances  between  the  pins  along  the  rulers,  that 


Art.  7.]  QUADRILATERALS.  105 

is,  AB  and  GD^  must  be  equal ;  also,  along  the  cross  pieces,  tliat  is,  A  G 
and  BD.  If  one  ruler  is  held  fast  while  the  other  is  moved,  lines  drawn 
along  the  edge  of  the  other  ruler,  at  different  positions,  are  parallel. 


8.  Exercises, — 1.  The  sum  of  two  opposite  sides  of  any  quadri- 
lateral circumscribed  about  a  circle  is  equal  to  the  sum  of  the  other  two 
sides.     [V,  18,  Ex.  3.] 

2.  The  two  angles  at  the  center  of  the  circle  which  are  subtended  by 
opposite  sides  of  the  circumscribed  quadrilateral  are  supplementary. 

3.  The  diagonals  of  a  rectangle  are  equal ;  and  conversely. 

4.  The  diagonals  of  a  rhombus  bisect  its  angles;  and  are  perpendicu- 
lar to  each  other ;  and  conversely. 

5.  If  the  four  midpoints  of  the  sides  of  any  quadrilateral  are  joined  by 
straight  lines,  those  lines  form  a  parallelogram.  [Make  the  diagonals  of 
the  quadrilateral,  and  see  V,  23.] 

6.  If  four  points  are  taken,  one  in  each 
side  of  a  square,  at  equal  distances  from 
the  four  vertices,  the  figure  formed  by 
joining  these  successive  points  is  a  square ; 
for  example,  the  points  A^  B,  G,  and  B. 

7.  If  the  same  thing  is  done  in  a  paral- 
lelogram, the  figure  formed  is  a  parallelo- 
gram. 

8.  The  lines  that  bisect  the  angles  of  a 
parallelogram  form  a  rectangle. 

9.  The  diagonals  of  the  rectangle  in  the  last  exercise  are  parallel  to 
the  sides  of  the  parallelogram,  and  are  midway  between  them.  [Produce 
each  bisector  till  it  reaches  both  the  other  sides  of  the  parallelogram,  one 
of  them  being  produced.] 

10.  When  one  diagonal  of  a  quadrilateral  divides  the  figure  into  equal 
triangles,  is  the  figure  necessarily  a  parallelogram  ? 

11.  If  two  equal  circumferences  intersect  at  right  angles,  the  common 
chord  is  equal  to  the  distance  between  the  centers.     [lY,  17,  i.] 

12.  If  equal  parallels  are  projected  on  the  same  line,  the  projections 
are  equal.    [Ill,  27.] 

13.  Parallels  that  have  equal  projections  on  the  same  line  are 
equal. 

14.  What  geometrical  principles  are  applied  in  making  a  win- 
dow frame  rectangular?  In  the  construction  and  use  of  parallel 
rulers? 


106 


PLANE  GEOMETRY. 
Measure  of  Area. 


[Chap.  VI. 


9,  The  standard  measure  of  surfaces  is  a  square.  That  is, 
the  unit  of  area  is  a  square,  having  for  its  side  the  unit  of 
length. 

It  is  used  because  of  its  simplicity.  It  has  the  same  length 
throughout  its  breadth,  and  the  same  breadth  throughout  its 
length  ;  and  the  length  and  breadth  are  equal. 


Area  of  Kectangles. 

Theorem. —  The  area  of  a  rectangle  is  equal  to  the  product 
of  its  base  and  altitude. 

That  is,  if  the  number  of  units  of  length  in  the  base 
is  multiplied  by  the  number  of  units  of  length  in  the  al- 
titude, the  product  is  the  number  of  units  of  area  in  the 
rectangle. 

Any  straight  line  may  be  the  unit  of  length.  The  square 
on  it  is  the  unit  of  area.  There  are  two  cases :  1st,  when 
the  base  and  the  altitude  of  the  rectangle  have  a  common 
measure  ;  and  2d,  when  these  lines  have  no  common  meas- 
ure. 

1st.  Divide  the  base  and  the  altitude  into  segments,  each 
equal  to  the  common  measure,  and  let  it  be  the  unit  of  length. 
Through  every  point  of  division  of  the  base  make  lines  parallel 
to  the  altitude,  and  through  every  point  of  division  of  the  alti- 
tude make  lines  parallel  to  the 
base.  The  lines  of  one  set  of 
parallels  are  perpendicular  to 
those  of  the  other,  and  all  the 
segments  are  equal  to  the  unit 
of  length.  Thus  the  rectangle 
is  divided  into  equal  squares, 
each  square  equal  to  the  unit 
of  area.     The  number  of  these 

squares  is  the  product  of  the  number  in  one  row  by  the  number 
of  rows,  that  is,  the  number  of  times  the  unit  of  length  is  in  the 
base  by  the  number  of  times  it  is  in  the  altitude. 


f t -j h • 

[ 1 |- 1- 

III! 

1 I I I 


Art.  9.]  QUADRILATERALS.  107 

Therefore  the  area  of  the  rectangle  is  to  the  unit  of  area  as 
the  product  of  the  base  and  altitude  of  the  rectangle  is  to  unity. 
This  may  be  expressed 

Here  h  and  a  are  the  numerical  ratios  of  the  base  and  altitude 
of  the  rectangle  to  the  unit  of  length.  In  this  1st  case,  h  and  a 
are  integers. 

2d.  When  the  base  and  the  altitude  of  the  rectangle  have  no 
common*  measure,  if  the  same  method  is  pursued  as  before,  a 
part  of  the  rectangle  may  be  measured  by  using  a  square  whose 
side  is  an  aliquot  part  of  the  altitude.  By  using  a  square  whose 
side  is  regularly  smaller  than  the  unmeasured  part  of  the  base  of 
the  rectangle,  the  part  of  the  rectangle  measured  is  an  increasing 
variable  rectangle,  and  its  limit  is  the  given  rectangle.  Also  the 
base  of  this  variable  rectangle  is  a  variable,  whose  limit  is  the 
given  base.  For  an  aliquot  part  of  the  altitude  may  be  taken 
(Postulate  of  Extent)  so  small  that  the  base  of  the  given 
rectangle  and  its  area  are  exhausted  within  less  than  any  assign- 
able difference.  At  every  step  the  variable  area  is  to  the  unit 
of  area  as  the  product  of  the  variable  base  and  the  altitude  is  to 
the  square  of  unity,  as  demonstrated  above.  Substituting  their 
limits  for  these  always  equal  variable  ratios,  M  '.  V  =  b  a  :  1. 
Hence, 

ji=zba  zr. 

Here  b  and  a  represent  the  ratios  of  the  base  and  altitude  to 
the  unit  of  length.  They  may  be  any  numbers  either  commen- 
surable or  incommensurable  with  unity. 

Area  of  Parallelograms. 

10.  Theorem. — The  area  of  a  parallelogram  is  equal  to  the 
product  of  its  base  and  altitude. 

At  the  ends  of  the  base  AB 
erect  perpendiculars,  and  produce 
them  till  they  meet  the  opposite 
side,  in  the  points  £J  and  JFl 

Now  the  right-angled  triangles 


IT     D 


108 


PLANE  GEOMETRY. 


[Chap.  VI. 


F C 

V 


AED  and  BFC  are  equal,  haviDg  the  side  BF  equal  to  AE, 

since  they  are  perpendiculars  between  parallels ;   and  the  side 

BC  equal  to  AD.      If  each   of 

these  equal  triangles  is  subtracted 

from  the   entire  figure,  ABGE, 

the     remainders     ABFE      and 

ABCD  must  be  equivalent.     But 

ABFE  is  a  rectangle  having  the 

same    base    and   altitude    as   the 

parallelogram    AB  CD.       Hence, 

the  area  of  the  parallelogram  is  equal  to  the  product  of  the  base 

and  altitude. 

Corollaries. — I.  Any  two  parallelograms  have  their  areas  in 
the  same  ratio  as  the  product  of  the  base  and  altitude  of  one  to 
the  product  of  the  base  and  altitude  of  the  other. 

II.  Parallelograms  of  equal  altitude  have  the  same  ratio  as 
their  bases,  and  parallelograms  of  equal  bases  have  the  same 
ratio  as  their  altitudes. 

III.  Two  parallelograms  are  equivalent  when  they  have 
equal  bases  and  equal  altitudes  ;  also,  when  the  dimensions  of 
the  one  are  the  extremes,  and  the  dimensions  of  the  other  are 
the  means,  of  a  proportion. 

The  base  and  altitude  are  the  dimensions  of  a  parallelogram. 
A  rectangle  is  said  to  be  contained  by  its  dimensions. 

IV.  If  two  chords  of  a  circle  cut  each  other,  the  rectangle 
contained  by  the  segments  of  one  is  equivalent  to  that  contained 
by  the  segments  of  the  other. 


Area  of  Triangles. 

11.   Theorem. — The  area  of  a  triangle  is  equal  to  half  the 
product  of  its  base  and  altitude. 

For  any  triangle  is  one  half  of 
a  parallelogram  having  the  same 
base  and  altitude. 

Corollaries. — I.  The  areas  of 
two  triangles  have  the  ratio  of  the 
product  of  the  base  and  altitude 
of  one  to  the  product  of  the  same  dimensions  of  the  other. 


i 


Art.  11.]  QUADRILATERALS.  109 

IL  Triangles  of  equal  altitudes  have  the  same  ratio  as  their 
bases  ;  and  triangles  of  equal  bases  have  the  same  ratio  as  their 
altitudes. 

III.  Two  triangles  are  equivalent  when  they  have  equal  bases 
and  equal  altitudes. 

IV.  If  a  parallelogram  and  a  triangle  have  equal  bases  and 
equal  altitudes,  the  area  of  the  parallelogram  is  double  that  of 
the  triangle. 

12.  Theorem, — If  from  half  the  sum  of  the  three  sides  of  a 
triangle  each  side  is  subtracted^  and  if  these  rem^ainders  and  the 
half  sum  are  multiplied  together y  then  the  square  root  of  the  pro- 
duct is  the  area  of  the  triangle. 

Let   a,  by  and  c  be  the 
sides  of  the  triangle,  h  the  5'^^^:  7^""^"J^ 

altitude,  and  m  and  n  the  x     n     \  m  — -^^ 

segments   of   the   base,  a  ;  ^ 

that  is,  these  letters  repre- 
sent the  ratios  of  these  lines  to  some  unit  of  length. 

Then  (V,  SI,  i)         m  =  «-l+|i=l£!. 

Since  7i,  m,  and  5  make  a  right-angled  triangle  (Y,  30), 

Substituting  for  m  its  value  from  the  preceding  equation, 
and  bearing  in  mind  that  the  difference  of  two  squares  is  equal 
to  the  product  of  the  sum  and  difference  of  the  roots,  we  have 

=  V  + 2^ )    V 2^ ) 

_  g"  +  2  g5  +  5"  —  c'       c''  —  g"  +  2  a5  —  5" 

~  2~«  ^  2  a 

=  {(i  +  ^)''  —  c^      c'  —  ja  —  by 

~  2a  ^  2a 

_  (a  +  h  +  c)  (a  +  i  —  c)  (a  +  c  —  h)  (b  +  c  —  a) 

"  4  a' 


110 


PLANE  GEOMETRY. 


[Chap.  VI. 


Taking  the  square  root  and  multiplying  by  ^  a, 
■^  ah  =  -^  a  Y  


a) 


4.0" 


l)(l>  +  c  —  a) 


=  V  tV  (a  +  &  +  c)  (a  +  &  —  c)  (a  +  c 

But  i  ah  is  the  measure 
of  the  area  of  the  triangle. 

The  expression  may  be 
reduced  by  letting  p  repre- 
sent the  perimeter.     Then, 

a  +  &  —  c  =  p  —  2c  =  2(|-j9  —  c), 
a  +  c  — &=^  — 2  5  =  2(1-^  — &), 
I  +  c  —  a=p  —  2a=2  {\p  —  a). 

Substituting  these  values,  we  have, 

area  =  V  ^pikp  —  a)  (ii?  —  &)  {^p  —  c). 

13.  Theorem. — The  areas  of  similar  triangles  are  to  each 
other  as  the  squares  of  homologous  lines. 


Let  AEI  and  B  CD  be  similar  triangles,  and  10  and  DH 
homologous  altitudes. 

Then,  IO'.DH=AE\BC. 

But,  \AE'.\BCz=AE'.BC. 

Multiplying, 


^AExIO-.^BCxBIT^AE  :  BC\ 


That  is. 


area  JL^J:  area  J5aZ>  =  ^^  :  BC, 


The  second  power  of  a  ratio  is  sometimes  called  the  dupli- 
cate of  it.  Thus,  the  superficial  ratio  of  similar  triangles  is  the 
duplicate  of  their  linear  ratio. 


Art.  14.]  QUADRILATERALS.  HI 

14.  Theorem, — If  two  triangles  have  an  angle  in  one  equal, 
to  an  angle  in  the  other,  their  areas  are  to  each  other  as  the  pro- 
ducts  of  the  sides  including  the  equal  angles. 

Place  the  triangles  AB  C 
and  ADF,  so  that  the  equal  ^ 

angles  may  coincide   at  A, 
Join  CD, 

Taking  AB  and  AD  as 
the  bases  of  the  triangles 
ABC  and  ADC,  we  have 
(11,11) 

ABC',ADC  =  AB'.AD. 
Likewise,  AD C  :  ADF  =AC:  AF, 

Multiplying,  and  canceling  the  factor  AD  6', 

ABC  ,ADF=  AB  ,AC  '.AD ,  AF, 


Area  of  Trapezoids. 

15.  Theorem. — The  area  of  a  trapezoid  is  equal  to  half  the 
product  of  its  altitude  by  the  sum  of  its  bases, 

'  The  trapezoid  may  be  divided  by  a  diagonal  into  two 
triangles,  each  having  for  its  base  one  base  of  the  trape- 
zoid. 

The  altitude  of  each  of  these  triangles  is  equal  to  that  of  the 
trapezoid.  The  area  of  each  triangle  being  half  the  product  of 
the  common  altitude  by  its  base,  the  area  of  their  sum,  or  of  the 
whole  trapezoid,  is  half  the  product  of  the  altitude  by  the  sum 
of  the  bases. 

Corollary. — The  area  of  ^  ^ 

a  trapezoid  is  equal  to  the 
product  of  its  altitude  by  a 
line  midway  between  the 
bases  and  parallel  to  them  ; 
for  FF  is  half  the  sum 
of    AB    and    CD,     FF   is    the    arithmetical    mean    of    the 


K, 


112  PLANE  GEOMETRY.  [Chap.  VI. 

16.  Applications. — Enlightened  nations  attach  great  importance 
to  exact  standard  measures. 

The  standards  generally  used  for  the  measure  of  surface  are  the 
squares  described  upon  linear  standards,  such  as  a  yard,  a  meter,  or  some 
other ;  but  the  acre,  the  common  unit  for  measuring  land  in  this  country, 
is  an  exception. 

The  public  lands  sold  by  the  United  States  are  divided  into  square 
townships,  each  containiag  thirty-six  square  miles,  called  sections,  but  in 
some  of  the  townships  there  are  only  twenty-five  square  miles. 

If  through  one  vertex  of  any  polygon,  however  irregular,  an  in- 
definite straight  line  is  drawn,  and  on  this  line  a  perpendicular  is  dropped 
from  every  other  vertex,  the  whole  is  divided  into  trapezoids  and  trian- 
gles. By  measuring  these,  the  entire  area  is  ascertained.  This  is  the 
usual  mode  of  surveying  land. 

17.  Exercises. — 1.  What  is  the  area  of  a  lot,  in  the  shape  of  a 
right-angled  triangle,  the  longest  side  being  100  yards,  and  one  of  the 
other  sides  36  yards? 

2.  Two  parallelograms  having  the  same  base  and  altitude  are  equiva- 
lent.    To  be  demonstrated  without  using  Articles  9, 10  or  11.     [V,  12.] 

3.  The  diagonals  of  a  parallelogram  divide  it  into  four  equivalent 
triangles. 

4.  Any  straight  line  through  the  midpoint  of  the  diagonals  of  a 
parallelogram  divides  it  into  equivalent  parts. 

5.  What  is  the  locus  of  the  vertices  of  those  triangles  that  have  equal 
areas  and  a  common  base  ? 

6.  The  area  of  a  triangle  is  equal  to  half  the  product  of  the  perimeter 
by  the  radius  of  the  inscribed  circle.  [Extend  a  line  from  the  center  to 
each  vertex  and  to  each  point  of  contact.] 

7.  A  triangle  is  divided  into  equivalent  parts  by  a  medial  line.  To 
be  demonstrated  without  using  any  theorem  in  this  chapter.  [From  the 
foot  of  the  medial  extend  lines  parallel  to  the  sides.] 

8.  If  0  is  any  point  in  a  parallelogram  ABCB,  the  triangles  OAB 
and  OCD  are  together  equivalent  to  half  of  the  parallelogram. 
[Art.  8.] 

9.  If  perpendiculars  fall  from  any  point  within  an  equilateral 
triangle  to  the  three  sides,  their  sum  is  equal  to  the  altitude  of 
the  triangle.  How  should  this  be  stated  when  the  point  is  outside 
of  til e  triangle? 

10.  If  two  triangles  have  two  sides  of  one  respectively  equal  to  two 
sides  of  the  other  and  the  included  angles  supplementary,  the  triangles 
are  equivalent.     [Ill,  7,  in.] 


Art.  18.]  QUADRILATERALS.  113 

The  Algebraic  Method. 

18«  In  the  last  chapter  and  in  this,  the  length  of  a  line  has 
been  represented  several  times  by  a  single  letter,  the  letter 
representing  the  number  of  times  some  unit  of  length  is  con- 
tained in  the  line,  that  is,  the  numerical  value  of  the  line.  The 
measure  of  area  is  always  a  product  of  two  such  numbers.  The 
area  of  a  square  is  the  second  power  of  its  length,  etc. 

Any  product  of  two  numerical  factors  may  be  taken  as 
representing  a  rectangular  area,  the  factors  being  the  two 
dimensions.  When  any  homogeneous  equation  of  the  second 
degree  is  known  to  be  universally  true,  one  form  of  this  truth  is 
a  principle  of  geometry.  We  owe  this  algebraic  method  of 
investigating  geometric  truth  to  the  mathematicians  of  the 
seventeenth  century. 

The  method  of  reasoning  directly  upon  the  geometrical  mag- 
nitudes was  pursued  by  the  ancients  twenty  centuries  before  ; 
and  it  is  usually  called,  by  way  of  distinction,  the  ancient 
method. 


19,  For  example  of  the  algebraic  method,  we  know  that 
whatever  are  the  values  of  a  and  h, 

(a +  5)2  =^2  +  52  -\-2ah. 

This  formula  includes  the  following 

Theorem. —  The  square  on  the  sum  of  two  straight  lines  is 
equivalent  to  the  sum  of  the  squares  on  the  two  lines,  increased 
by  twice  the  rectangle  contained  by  those  lines. 

This  may  be  demonstrated  by  the  ancient  method,  as  fol- 
lows : 

The  sum  of  two  lines  CD  and 
I>JE  is  CK    Upon   GE  erect  the  K_ 

square  JEK\  upon  CD  erect  the 
square  DII\  and  produce  the  sides 
of  this  second  square  to  iV  and 
P. 

Then  HK,  being  the  difference 
between  (7^  and  CH,  is  equal  to 
6 


/v 


JJ 


114 


PLANE  GEOMETRY, 


[Chap.  VL 


DE.  PG  and  FK  are  also  equal 
to  I>Ey  being  parallels  between 
parallels.  Likewise,  MG  and  GD 
are  each  equal  to  CD,  Therefore, 
the  square  on  C£J,  the  sum,  is 
composed  of  the  squares  on  CD 
and  on  DE,  and  of  two  rectangles, 
each  contained  by  these  two  lines. 


ff 


Ti 


N- 


E 


20.  Theorem. — The  square  on  the  difference  of  two  straight 
lines  is  equivalent  to  the  sum  of  the 
squares  on  the  two  lines,  diminished 
hy  twice  the  rectangle  contained  hy       ^ 
those  lines. 

This   is  a  consequence   of   the 
truth  of  the  equation, 


{a  —  hy  =  a^—2ab-\-bK 


0.-6 


The    student    may    demonstrate    it    also    by    the    ancient 
method. 


21.  Theorem. — The  rectangle  contained  hy  the  sum  and  the 
difference  of  two  straight  lines  is  equivalent  to  the  difference  of 
the  squares  on  those  lines. 

This  is  proved  by  the  principle 
expressed  in  the  equation, 

{a-\-h)  (a  —  b)  =  a"-—bK 

It  may  also  be  demonstrated  by 
aid  of  this  diagram. 


22.  Theorem. — The  square  on  the  side  that  subtends  an 
acute  angle  of  a  triangle  is  equivalent  to  the  sum  of  the  squares 
on  the  other  two  sides,  diminished  by  twice  the  rectangle  con- 
tained by  one  of  those  sides  and  the  projection  of  the  other  on 
that  side. 


Art.  22,] 


Q  UADRILA  TERAL8, 


115 


Let  a,  hf  and  c  be  the  sides  of  the  triangle,  h  the  base,  the 
angle  opposite  to  a  being  acute,  m  and  n  respectively  the  pro- 


jections of  a  and  of  c  on  5,  and  h  the  altitude.     Then  (Y,  30, 
and  VI,  20), 

aS  =  A2  +  m2 


23.  Theorem, —  7%e  square  on  the  side  that  subtends  an 
obtuse  angle  of  a  triangle  is  equivalent  to  the  mini  of  the  squares 
on  the  other  two  sides,  increased  by  twice  the  rectangle  of  one  of 
those  sides  and  the  projection  of  the  other  on  that  side. 

Representing  the  lines  as 
before,  the  angle  opposite  to 
a  being  obtuse, 

a3  =  A2  4-  m2 

=  A2  _|_  ^2  _|.  52  J^2nb 
=  c2  + J2  _|_2n5. 

Corollary. — If  the  square 
on  one  side  of  a  triangle  is 

equivalent  to  the  sum  of  the  squares  on  the  other  two  sides,  then 
the  opposite  angle  is  a  right  angle.  For  it  can  be  neither  acute 
nor  obtuse. 


116 


PLANE  GEOMETRY, 


[Chap.  VI. 


34.  Theorem. — Selecting  one  side  of  a  triangle  as  the  base, 
the  rectangle  contained  by  the  other  two  sides  is  equivalent  to  the 
rectangle  contained  by  the  altitude  of  the  triangle  and  the  diam- 
eter of  the  circumscribed  circle. 

Taking  BC  2.9,  the  base  of  the 
triangle,  AD  is  the  altitude  and 
AE  the  diameter  of  the  circum- 
scribed circle.     Join  EC. 

The  triangles  ABB  and  AEG 
have  the  angles  B  and  E  equal 
(IV,  22,  11),  also  the  angles  A  CE 
and  ABB.     Therefore, 

AC'.AB  =  AE'.AB. 

The  product  of  the  means  being  equal  to  that  of  the  extremes, 
the  proposition  is  proved  (10,  in). 


25.  Theorem. —  The  area  of  a  triangle  is  measured  by  the 
product  of  the  three  sides  multiplied  together ,  divided  by  four 
times  the  radius  of  the  circumscribed  circle. 

Let  b  represent  the  base,  a  and  c  the  sides,  h  the  altitude, 
and  r  the  radius.     Then,  as  just  proved. 


Therefore, 


2rh  =  ac. 
h  h  _ab  c 


26.  Theorem. — If  a  quadrilateral  is  inscribed  in  a  circUy 
the  sum  of  the  two  rectangles,  each  of  which  is  contained  by  op- 
posite sides  of  the  quadrilateral, 
is  equivalent  to  the  rectangle  con- 
tained by  its  diagonals. 

Make  the  angle  ABE  equal  to 
BBC.  Then  ABB  is  equal  to 
EBC.  Since  BBA  and  BCE 
are  equal  angles,  the  triangles 
ABB  and  EB  C  are  similar,  and 

BG:CE=BB'.BA, 


Art.  26.]  QUADRILATERALS.  117 

Likewise,  from  the  similar  triangles  BEA  and  B  CD, 

CI):EA  =  BD:AB, 

From  these  proportions  we  have  B  C '  DA  =  BD  *  CE  and 
CD  •  AB  =  BD '  EA,     By  addition, 

BC'DA  -{-CD'AB  =  BD'  CA. 

This  is  the  Ptolemaic  Theorem.      It  is  found  in  the 
Almagest,  written  by  Claudius  Ptolemy  in  the  second  century. 


27.  Theorem. — A  line  bisecting  the  vertical  angle  of  a  tri- 
angle divides  the  base  in  the 
ratio  of  the  other  two  sides. 

Taking  AB  and  BC  slb  the 
bases  of  the  two  triangles 
ABD  and  CBD,  they  have 
equal  altitudes  (III,  13).  There- 
fore (11,  ii), 

area  ABD  :  area  CBD  =  AB  :  CB. 

Regarding  AD  and  DC  sls  the  bases, 

area  ABD  :  area  CBD  =  AD  :  CD. 
Therefore,  AD  :  CD  =  AB  :  CB. 

Corollaries, — I.  If  the  ex- 
terior angle  at  the  vertex  is 
bisected,  the  base  is  divided 
externally,  in  the  same  ratio. 
The  words  of  the  above  demon- 
stration apply  to  this. 

II.  Representing  the  sides 
by  a  and  c  and  the  base  by  b,  the  segments  of  the  base  when 


divided  internally  are 
externally,  they  are  — 


a  -{-  G 
a 


b  and 


-  b  and  — 
c  a 


a-\-G 
c 


b.     When  it  is  divided 


5. 


118 


PLANE  GEOMETRY. 


[Chap.  VI. 


III.  If  the  angle  at  the  base  and  the  adjacent  exterior  angle 
are  both  bisected,  the  base  is  divided  harmonically  (III,  2). 

IV.  The  locus  of  the  points,  the  distances  of  each  of  which 
from  two  fixed  points,  A  and  G,  are  in  a  given  ratio,  is  a  cir- 
cumference cutting  the  line  AC  in  the  required  ratio  both 
internally  and  externally,  and  having  for  its  diameter  the  dis- 
tance between  the  points  of  intersection. 


The  Pythagorean  Theorem. 

28.  Since  numerical  equations  represent  geometrical  truths, 
the  following  theorem  might  be  inferred  from  Article  30,  Chap- 
ter V. 

This  theorem,  discovered  by  Pythagoras,  is  known  as  the 
Forty-seventh  Proposition,  that  being  its  number  in  the  First 
Book  of  Euclid's  Elements. 

It  has  been  demonstrated  in  a  great  variety  of  ways.  The 
first  demonstration  following  is  from  Euclid. 

Theorem.— The  square  on  the  hypotenuse  of  a  right-angled 
triangle  is  equivalent  to  the  sum  of  the  squares  on  the  sides 
that  contain  the  right  angle. 

Let  AB  C  be  a  right-angled  triangle,  having  the  right  angle 
JBA  C.  It  is  to  be  proved  that  the  square  on  BC  is  equivalent 
to  the  squares  on  BA  and  A  C.  Through  A,  make  AL  parallel 
to  BJD,  and  join  AB  and  FC. 

Then,  because  each  of  the 
angles  BAC  and  BAG  is 
right,  their  sum  is  two  right 
angles,  and  GA  C  is  one  straight 
line.  For  the  same  reason, 
BAH  is  one  straight  line. 

The  angles  FB  G  and  BBA 
are  equal,  since  each  is  the  sum 
of  a  right  angle  and  the  angle 
ABG.  The  two  triangles  JF!Z?  (7 
and  BBA   are   equal,  for  the 

side  FB  is  equal  to  BAy  and  the  side  BG  h  equal  to  BB,  and 
the  included  angles  are  equal,  as  just  proved. 


Art.  28.]  QUADRILATERALS.  119 

Kow,  the  area  of  the  parallelogram  BL  is  double  that  of  the 
triangle  DBA,  because  they  have  the  same  base  BB  and  the 
same  altitude  BL ;  and  the  area  of  the  square  BG  is  double 
that  of  the  triangle  BB  C^  because  they  have  the  same  base  BB^ 
and  the  same  altitude  jP6r.  But  doubles  of  equals  are  equal. 
Therefore,  the  parallelogram  BB  and  the  square  B  G  are  equiva- 
lent. 

In  the  same  manner,  by  joining  ABJ  and  BB,  it  is  demon- 
strated that  the  parallelogram  CB  and  the  square  CBT  are  equiva- 
lent.    Therefore,  the  whole  square  BB,  on  the  hypotenuse,  is 
equivalent  to  the  squares  BG  and 
Cir,  on  the  other  two  sides  of  the 
right-angled  triangle.  /T\^ 

Another  mode  of  demonstration  /   i     ,  n. 

is  by  dividing  the  three  squares  into  /  ^     |  N. 

parts,  such  that  the  several  parts  of  /.  | 

the    large    square  ^re    respectively  l^v        I  / 

equal  to  the  several  parts  of  the  two  i    ,  /\^  \      /  ' 

others.      The    diagram    shows    the  J .^bsZ. 

construction. 


29.  Exercises.— 1.  What  kind  of  a  triangle  is  that  whose  sides 
are  3,  4,  and  5  ?  That  whose  sides  are  14,  15,  and  20  ?  14,  15,  and  5  ? 
12,  13,  and  5  ? 

2.  What  is  the  radius  of  the  circle  inscribed  in  the  triangle  whose 
sides  are  8,  10,  and  12  ?    [Find  the  area  of  the  triangle.] 

3.  The  sum  of  the  areas  of  the  squares  on  the  sides  of  a  parallelogram 
is  equal  to  the  sum  of  the  areas  of  the  squares  on  its  diagonals.  [22  and 
23.] 

4.  If  m  and  n  are  any  numbers,  and  a  triangle  has  sides  equal  to  m^ 
+  n^,  m^  — n^,  and  2mn  units  respectively,  the  triangle  is  right-angled. 

5.  In  a  circle  of  26  inches  diameter,  a  chord  is  distant  one  foot  from 
the  center;  how  long  is  the  chord?     [lY,  11,  i.] 

6.  If  two  equal  chords  of  a  circle  cut  each  other,  their  segments  are 
respectively  equal.  [Find  the  value  of  a  segment  of  one  in  terms  of  a 
segment  of  the  other.] 

7.  If  two  chords  of  a  circle  cut  each  other  at  right  angles,  the  sum  of 
the  squares  on  the  four  segments  is  equivalent  to  the  square  on  the  diam- 
eter. 


120  PLANE  GEOMETRY.  [Chap.  VI. 

8.  If,  from  any  point  in  a  square,  lines  are  made  to  the  four  vertices, 
also  perpendiculars  to  the  four  sides,  the  sum  of  the  areas  of  the  squares 
on  the  first  four  lines  is  double  the  sura  of  the  areas  of  the  squares  on  the 
perpendiculars.     [28.] 

9.  If  a  straight  line  is  divided  internally  or  externally,  the  sum  of  the 
areas  of  the  squares  on  the  segments  is  double  the  sum  of  the  areas  of  the 
squares  on  half  the  line  and  on  the  line  between  the  point  of  division  and 
the  midpoint  of  the  given  line. 

10.  Given  the  lengths  of  the  bases  and  of  the  altitude  of  a  trapezoid, 
to  find  the  distance  from  either  base  to  the  point  of  meeting  of  the  oblique 
sides  produced.     [V,  22,  i.] 

11.  When  the  base  of  a  triangle  is  given,  also  the  difference  of  the 
squares  of  the  sides,  the  locus  of  the  vertex  is  a  straight  line  perpendicu- 
lar to  the  base.    [Y,  31.] 


30.  Scholium. — The  study  of  triangles  developed  the  doc- 
trine of  similarity.  The  principles  of  mensuration  of  surfaces 
and  the  theory  of  equivalent  figures  are  developed  from  the 
properties  of  quadrilaterals. 

The  algebraic  method,  applied  to  investigations  in  the  higher 
mathematics,  has  led  to  the  greatest  achievements  in  the 
science. 

The  Pythagorean  Theorem  and  the  theory  of  similar  trian- 
gles are  the  basis  of  Trigonometry. 


31,  Miscellaneous  Exercises. — 1.  If  two  triangles  have  a 
common  base,  but  are  on  opposite  sides  of  it,  the  line  Joining  their  ver- 
tices is  cut  by  the  base  in  the  ratio  of  the  areas  of  the  triangle. 
[11,  II.] 

2.  In  any  parallelogram,  the  distance  of  one  vertex  from  a  straight 
line  passing  through  the  opposite  vertex  is  equal  to  the  sum  or  difference 
of  the  distances  of  the  line  from  the  other  two  vertices,  according  as  the 
line  is  without  or  within  the  parallelogram.     [3.] 

3.  From  any  point  in  the  base  of  a  triangle,  lines  are  made  parallel  to 
the  two  sides:  find  the  locus  of  the  centers  of  symmetry  of  the  parallelo- 
grams so  formed.     [6.] 

4.  If,  from  the  sides  of  the  triangle  ABC^  AD^  BE,  and  CF  are  cut 
off,  each  equal  to  two  thirds  of  the  side  from  which  it  is  cut,  then  the 
area  of  the  triangle  DEF  is  one  third  that  of  ABG.     [14.] 

5.  Find  the  locus  of  the  points  such  that  the  sum  of  the  squares  of 


Art.  31.]  QUADRILATERALS.  121 

the  distances  of  eacli  from  two  given  points  is  equivalent  to  the  square  of 
the  line  joining  the  given  points.     [28.] 

6.  If  a  triangle  is  equilateral,  the  radii  of  the  inscribed,  circumscribed, 
and  escribed  circles  are  in  the  ratio  of  1,  2,  3.     [V,  43,  Ex.  11.] 


Problems  in  Drawing. 

33,  The  Squaring  or  Quadrature  of  a  figure  consists  in 
drawing  a  square  equivalent  to  it. 

I. — To  draw  a  polygon  equivalent  to  a  given  one,  hut  with  a  less  num- 
ter  of  sides. 

Let  ABODE  be  the  given  polygon.  Join  DA.  Produce  BA,  and 
through  E  draw  ^F  parallel  to  DA.    Join  DF. 

The  triangles  DAF  and 
DAE  are  equivalent,  for  they 
have  the  same  base  DA,  and 
their  vertices  are  in  the  line  EF 
parallel  to  the  base.  To  each 
of  these  equals  add  the  figure 
ABGD,  and  we  have  the  quad- 
rilateral FBGD  equivalent  to 
the  polygon  ABODE.  In  this 
manner,   the   number    of   sides 

may  be  diminished  till  a  triangle  is  formed  equivalent  to  the  given  poly- 
gon.   In  this  diagram  it  is  the  triangle  FDG. 

II. — To  represent  \/ab  geometrically.   ^ 

The  square  root  of  the  product  of  two  numbers  is  their  mean  propor- 
tional. The  problem  assumes  that  the  letters  represent  lines.  Therefore, 
find  a  mean  proportional  between  two  lines  of  the  lengths  a  and  &  (V, 
44,  ix). 

Geometrical  representation  of  algebraic  formulas  is  used  in 
the  solution  of  problems  in  drawing.  When  a  relation  between 
the  magnitudes  given  and  the  magnitudes  required  can  be  stated 
as  an  equation,  the  algebraic  solution  may  be  made  first.  All 
that  remains  is  to  represent  the  result  geometrically. 

III. — To  draw  a  rectangle  with  a  given  base,  equivalent  to  a  given  par- 
allelogram. 

The  altitude  of  the  rectangle  is  to  be  found.  If  c  is  the  given  base, 
and  &  and  a  the  dimensions  of  the  given  parallelogram,  let  x  represent 
the  required  altitude.    Then  c :  5  =  a :  a;  (V,  44,  v). 


122  PLANE  GEOMETRY.  [Chap.  VL 

ly. — To  draw  a  square  equivalent  to  a  given  parallelogram. 
If  &  and  a  are  the  dimensions  of  the  parallelogram,  let  x  represent  the 
side  of  the  square.     Then  x^  =  5a.     Therefore  a;  =  Via. 
Y. — To  draw  a  square  equivalent  to  a  given  triangle. 
aj*  =  i &a.    Therefore  ^l):x  =  x:a. 


33.  Exercises  in  Drawing. — 1.  To  divide  a  given  triangle  into 
any  number  of  equivalent  triangles. 

2.  To  divide  a  given  trapezoid  into  any  number  of  equivalent  trape- 
zoids. 

3.  To  divide  a  straight  line  into  two  segments  such  that  the  area  of 
the  square  on  one  is  four  times  that  of  the  square  on  the  otlier. 

4.  Represent  this  formula  by  a  diagram:  (a  +  5  +  c)^  =  a^  +  52  + 
c^  +  2  a&  +  2  ac  +  2  &c. 

5.  To  represent  \^{a  +  l)  {a  —  I)  geometrically. 

6.  To  draw  a  square  equivalent  to  two  given  squares.     To  three. 

7.  To  divide  any  two  squares  into  parts  that  may  be  combined  to 
form  one  square. 

8.  To  draw  a  square  with  a  given  straight  line  as  its  diagonal. 

9.  To  draw  a  triangle,  having  the  base  and  one  of  the  angles  at  the 
base  given,  and  an  area  equal  to  that  of  a  given  square. 

10.  To  draw  a  parallelogram  equivalent  to  a  given  triangle,  and  having 
an  angle  equal  to  a  given  angle. 

11.  From  a  given  isosceles  triangle,  to  cut  off  a  trapezoid  having  the 
same  base  as  the  triangle,  and  the  remaining  three  sides  equal  to  each 
other. 

12.  To  divide  a  triangle  into  two 
equivalent  parts,  by  a  line  drawn  from 
a  given  point  in  one  of  the  sides ;  viz., 
to  divide  the  triangle  ABG  into  equiva- 
lent parts  by  a  line  from  D. 

13.  To  divide  a  triangle  into  three 

equivalent  triangles  by  lines  that  meet  at  a  point  within  the  given  tri- 
angle. 

14.  Can  every  triangle  be  divided  into  two  equal  parts?  Into  three? 
Into  nine  ? 


CHAPTER  YII. 
POLYGONS. 

Article  1. — Polygons  have  been  defined  and  classified 
according  to  the  number  of  sides  (II,  15). 

A  Convex  Polygon  has  all  its  diagonals  interior. 

A  Concave  Polygon  has 
at  least  one  diagonal  exterior. 
If  the  angles  are  reojarded  as 
toward  the  figure,  a  concave 
polygon  must  have  a  reflex 
angle. 

A   Regular    Polygon   is 
both  equilateral  and  equiangular.     The   square   and  the   equi- 
lateral triangle  are  regular  polygons. 

Diagonals. 

2.  Theorem. —  The  number  of  diagonals  from  any  vertex  of 
a  polygon  is  three  less  than  the  number  of  sides. 

For,  from  any  vertex,  a  diagonal  may  extend  to  every  other 
vertex  except  one  on  each  side. 

Corollaries. — I.  The  diagonals  from  one  vertex  divide  a 
polygon  into  as  many  triangles  as  the  polygon  has  sides,  less 
two. 

A  polygon  may  be  divided  into  a  greater  number  of  trian- 
gles, in  various  ways  ;  but  not  into  a  less  number  than  here 
stated. 

II.  The  whole  number  of  diagonals  possible  in  a  polygon  of 
n  sides,  is  i  ti  (w  —  3).     For,  if  we  count  the  diagonals  at  all  the 


124  PLANE  GEOMETRY.  [Chap.  VII. 

n  vertices,  we  have  n  {n  —  3),  but  this  is  counting  each  diagonal 
at  both  ends. 

Sum  of  the  Angles. 

3.  Theorem. — The  sum  of  all  the  angles  of  a  polygon  is 
equal  to  twice  as  many  right  angles  as  the  polygon  has  sides, 
less  two. 

For  the  polygon  may  be  divided  into  as  many  triangles  as  it 
has  sides,  less  two ;  and  the  angles  of  these  triangles  coincide 
altogether  with  those  of  the  polygon. 

The  sum  of  the  angles  of  each  triangle  is  two  right  angles. 
Therefore,  the  sum  of  the  angles  of  the  polygon  is  equal  to  twice 
as  many  right  angles  as  it  has  sides,  less  two. 

In  applying  this  theorem  to  a  concave  figure,  the  value  of 
the  re-entrant  angle  must  be  taken  on  the  side  toward  the  poly- 
gon, and  therefore  as  reflex. 

Let  B,  represent  a  right  angle  ;  then  the  sum  of  the  angles  of 
a  polygon  of  n  sides  is  2  (/i  —  S)  i2  ;  or,  it  may  be  written  thus, 
{2n  —  4)  72  ;  that  is  {n  —  2)  straight  angles. 

4.  Theorem:. — If  each  side  of  a  convex  polygon  is  produced, 
the  sum  of  all  the  exterior  angles  is  equal  to  four  right  angles. 

Let  the  sides  be  produced  either  all  to  the  right  or  all  to  the 
left.  Then,  from  any  point  in  the  plane,  extend  lines  parallel  to 
the  sides  thus  produced,  and  in  the  same  directions. 

The  angles  thus  formed  are  equal  in  number  to  the  exterior 
angles  of  the  polygon, 
and    are    respectively 
equal    to    them    (III, 
21).     But  the  sum  of  ^/\ 

those  formed  about  the  /      ^v?-  \    ' 

point  is  equal  to  four  /  j  ^V\^. 

right  angles.  /'    \  1 

Therefore,  the  sum  V^^;- — — — J. « 

of  the  exterior  angles  \ 

of  the  polygon  is  equal 
to  four  right  angles. 


i") 


Art.  4.]  POLYGONS,  135 

This  is  also  true  of  concave  polygons,  if  the  angle  formed  by 
producing  one  side  of  the  re-entrant 
angle  is  taken  negatively.  Thus, 
the  sum  of  the  exterior  angles  at  -S, 
(7,  JD,  E,  F,  and  G,  less  the  angle 
j5j  is  four  right  angles. 

In  going  from  B,  around  the 
polygon,  in  the  order  of  the  letters, 
every  exterior  angle  is  a  divergence, 

or  difference  of  direction,  to  the  right ;  but  at  H  the  divergence 
is  to  the  left.  Subtracting  H  from  the  sum  of  the  others,  the 
result  is  four  right  angles. 

Equal  Polygons, 

5.  Theorem. — Two  polygons  are  equal  when  they  are  com- 
posed of  the  same  number  of  triangles  respectively  equal  and 
similarly  arranged. 

This  is  an  immediate  consequence  of  the  definition  of  equal- 
ity  (II,  4). 

Corollary. — Conversely,  two  equal  polygons  may  be  divided 
into  the  same  number  of  triangles  respectively  equal  and  simi- 
larly arranged. 

6.  Theorem. — Two  polygons  are  equal  when  all  the  sides 
and  all  the  diagonals  from  o?ie  vertex  of  one  are  respectively 
equal  to  the  same  lines  in  the  other  and  are  similarly  arranged. 

For  each  triangle  in  one,  having  its  sides  respectively  equal 
to  those  of  the  similarly  situated  triangle  in  the  other  polygon, 
is  equal  to  it. 

Corollary. — Two  quadrilaterals  are  equal  when  the  four 
sides  and  a  diagonal  of  one  are  respectively  equal  to  the  four 
sides  and  the  same  diagonal  of  the  other. 

7.  Theorem. — Two  polygons  are  equal  when  all  the  sides 
and  angles  of  one  are  respectively  equal  to  the  same  elements  of 
the  other  and  are  similarly  arranged. 

For  each  triangle   in   one   is   equal  to  its  homologous  tri- 


126 


PLANE  GEOMETRY. 


[Chap.  VII. 


angle  in  the  other,  since  they  have  two  sides  and  the  included 
angle  equal. 


It  is  enough  for  the  hypothesis  of  this  theorem,  that  all  the 
angles  except  three  are  equal. 

Corollaries. — I.  Two  quadrilaterals  are  equal  when  the  four 
sides  and  an  angle  of  one  are  respectively  equal  to  the  four  sides 
and  the  similarly  situated  angle  of  the  other. 

II.  Two  parallelograms  are  equal  when  two  adjacent  sides 
and  the  included  angle  of  one  are  respectively  equal  to  those 
elements  of  the  other. 

III.  Two  rectangles  are  equal  when  two  adjacent  sides  of 
one  are  respectively  equal  to  those  elements  of  the  other. 

IV.  Two  squares  are  equal  when  a  side  of  one  is  equal  to  a 
side  of  the  other. 


8.  Exercises. — 1.  "What  is  the  number  of  diagonals  that  can  be  in 
a  pentagon  ? 

2.  How  many  sides  has  that  polygon  the  number  of  whose  diagonals 
is  seven  times  the  number  of  sides? 

3.  "Wiiat  is  the  sum  of  the  angles  of  a  hexagon?  Of  a  dodeca- 
gon? 

4.  A  convex  polygon  can  not  have  more  than  three  acute  an- 
gles. 

5.  Join  any  point  witbin  a  given  polygon  with  every  vertex  of  the 
polygon,  and  with  the  figure  thus  formed  demonstrate  the  theorem, 
Article  3. 

6.  Demonstrate  Article  4  by  means  of  Article  3. 


Akt.  9.]  POLYGONS.  127 

Similar  Polygons. 

9.  Theorem. — Similar  polygons  are  composed  of  the  same 
number  of  triangles,  respectively  similar  and  similarly  arranged. 

Since  the  figures  are  similar,  every  angle  in  one  has  its  cor- 
responding equal  angle  in  the  other.     If  homologous  diagonals 


divide  the  polygons  into  triangles,  every  angle  formed  has  its 
corresponding  equal  angle.  Therefore,  the  triangles  are  respect- 
ively similar  and  are  similarly  arranged. 

Corollary. — Two  similar  polygons  may  be  so  placed  that 
their  homologous  sides  are  parallel.  Then  there  is  a  center  of 
similarity  as  in  the  case  of  triangles  (V,  27  and  28),  which  is 
external  when  the  corresponding  sides  are  arranged  in  the  same 
order,  and  interaal  when  they  are  in  the  reverse  order. 

10.  Theorem. — If  two  polygons  are  composed  of  the  same 
number  of  triangles  respectively  similar  and  similarly  arranged, 
the  polygons  are  similar. 

By  the  hypothesis,  all  the  angles  formed  by  the  given  lines 
in  one  polygon  have  their  corresponding  equal  angles  in  the 
other.  It  remains  to  be  proved  that  any  other  angle  formed  by 
lines  in  one  has  its  corresponding  equal  angle  in  the  other  poly- 
gon. 

Whatever  line  is  made  in  one  polygon,  a  homologous  line 
may  be  made  in  the  corresponding  similar  triangle  or  triangles 
of  the  other  polygon.  Since  the  triangles  are  similar,  every 
angle  made  by  one  of  these  lines  has  its  corresponding  equal 
angle  in  the  other  polygon. 


128  PLANE  GEOMETRY.  [Chap.  VII. 

11,  Theorem. — Two  polygons  are  similar  when  the  angles 
formed  by  the  sides  are  respectively  equal,  and  there  is  the  same 
ratio  between  each  side  of  one  and  its  homologous  side  of  the 
other  polygon. 

Let  all  the  diagonals  possible  extend  from  a  vertex  A  of  one 
polygon,  and  the  same  from  the  homologous  vertex  B  of  the 
other  polygon. 


The  triangles  AEI  and  B  CD  are  similar,  because  they  have 
two  sides  proportional,  and  the  included  angles  equal. 

Therefore,  EI :  CD  =  AI :  BD. 

But,  by  hypothesis, 

EI\CD  =  IO\DF. 
Then,  AI.BjD  =  IO'.  DF. 

Subtract  the  equal  angles  EIA  and  CBB  from  the  equal 
angles  EIO  and  CDF\  the  remainders  AIO  and  BDF  are 
equal.  Hence,  the  triangles  AIO  and  BDF  are  similar.  In 
the  same  manner,  every  triangle  of  the  first  polygon  is  similar  to 
its  corresponding  triangle  in  the  other.  Therefore,  the  figures 
are  similar. 

As  in  the  case  of  equal  polygons  (7),  it  is  only  necessary  to 
the  hypothesis  of  this  proposition  that  all  the  angles  except 
three  in  one  polygon  be  equal  to  the  homologous  angles  in  the 
other. 

1 2 .  Theorem.. — In  similar  polygons  the  ratio  of  two  hoinolo- 
gous  lines  is  the  same  as  of  any  other  two  homologous  lines. 


Art.  12.] 


POLYGONS. 


129 


For,  since  the  polygons  are  similar,  tlie  triangles  which  com- 
pose them  are  also  similar,  and 


AE\BC  =  EI',  CD  =  AI:  J3D  =  10  :  DF,  etc. 

This  common  ratio  is  the  linear  ratio  of  the  two  figures. 
Corollary. — The  perimeters  of  similar  polygons  are  to  each 
other  as  any  two  homologous  lines. 

13,  Theorem. — The  area  of  any  polygon  is  to  the  area  of 
a  similar  polygon  as  the  square  on  any  line  of  the  first  is  to  the 
square  on  the  homologous  line  of  the  second. 

Let  the  polygons  BCD,  FGH,  and  AEI,  OUT,  be  divided 
into  triangles  by  homologous  diagonals.  The  triangles  are  re- 
spectively similar. 


8  2 

Therefore  (YI,  13),  area  BCD  :  area  AEI  —  BD  I'AI  = 
area  BDE :  area  AIO  =  BE : 'AO  =  area  BEG  :  area  AOV 


=  BG'.AU=  area  BGH:  area  A  UY. 

Selecting  from  these  equal  ratios  the  triangles,  area  B  CD  : 
area  AEI  =  area  BDE :  area  AIO  =  area  BEG  :  area  AOZT 
=  area  B  GIT :  area  A  TIY. 


130  PLANE  GEOMETRY.  [Chap.  VII. 

Therefore  (I,  3,  vi),  area  BCDFGHB  :  area  AEIOUTA 

8     Z 

=  area  B  CD  :  area  A£JI ;  or,  as  ^  (7 :  AF  ;  or,  as  the  areas  of 
any  other  homologous  parts  ;  or,  as  the  squares  of  any  other 
homologous  lines. 

Corollary, — The  superficial  ratio  of  two  similar  polygons  is 
the  second  power  or  duplicate  of  their  linear  ratio  ;  and  con- 
versely, the  linear  ratio  is  the  square  root  of  the  superficial  ratio. 

14,  Exercises. — 1.  Compose  two  polygons  of  the  same  number 
of  triangles  respectively  similar,- but  not  similarly  arranged. 

2.  What  is  the  relation  between  the  areas  of  the  equilateral  triangles 
described  on  the  three  sides  of  a  right-angled  triangle? 

3.  Two  parallelograms  are  similar  when  they  have  an  angle  in  one 
equal  to  an  angle  in  the  other,  and  these  equal  angles  included  between 
proportional  sides. 

4.  Two  irregular  garden  plats,  of  the  same  shape,  contain,  respect- 
ively, 18  and  32  square  yards:  required 

their  linear  ratio. 

5.  If,  through  any  point  in  the  diag- 
onal of  a  parallelogram,  lines  are  made 
parallel  to  the  sides,  four  parallelograms 
are  formed ;  the  two  through  which  the 
diagonal  does  not  pass  are  equivalent. 

These  two  are  called  the  complements  of  the  parallelograms  about  the 
diagonal.    Thus  a  and  5  are  the  complements. 

Regular  Polygons. 

15.  Theorem. — Lines  bisecting  the  several  angles  of  a  regu- 
lar polygon  all  meet  in  one  point ;  this  point  is  equally  distant 
from  all  the  vertices,  and  is  also  equally  distant  from  all  the 
sides  of  the  polygon. 

Every  triangle  formed  by  one 
side  of  the  polygon  and  two  con-  ^  b 

secutive  bisecting  lines  is  isosceles 
(V,  6),  and  all  the  triangles  so 
formed  are  equal  (V,  12).  Hence, 
BO  is  cut  by  the  lines  AO  and 
(70  at  the  same  point.     Thus  all 


Art.  15.]  POLYGONS.  131 

the  bisecting  lines  meet  at  one  point  which  is  equally  distant 
from  the  vertices. 

These  equal  isosceles  triangles  must  have  equal  altitudes,  that 
is,  the  point  0  is  equally  distant  from  all  the  sides  of  the  poly- 
gon. 

Corollaries. — I.  Every  regular  polygon  may  have  a  circle 
circumscribed  about  it. 

II.  Every  regular  polygon  may  have  a  circle  inscribed  in 
it. 

III.  An  angle  formed  at  the  center  of  a  regular  polygon  by 
lines  from  adjacent  vertices  is  an  aliquot  part  of  four  right 
angles,  being  the  quotient  of  four  right  angles  divided  by  the 
number  of  the  sides  of  the  polygon. 

IV.  Every  regular  polygon  may  be  divided  into  as  many 
equal  isosceles  triangles  as  the  polygon  has  sides. 

V.  Conversely,  a  polygon  composed  of  equal  isosceles  tri- 
angles having  their  vertices  at  a  common  point  is  regular. 

The  Center  of  a  Regular  Polygon  is  the  point  equally 
distant  from  the  vertices.  The  Radius  is  the  distance  from  the 
center  to  a  vertex.  The  Apothem  is  the  distance  from  the 
center  to  a  side. 

VI.  If  a  regular  polygon  has  an  even  number  of  sides,  a  line 
from  one  vertex  through  the  center,  being  produced,  passes 
through  the  opposite  vertex,  and  is  an  axis  of  symmetry. 

VII.  If  a  regular  polygon  has  an  odd  number  of  sides, 
a  line  from  one  vertex  through  the  center,  being  produced, 
bisects  the  opposite  side  at  right  angles,  and  is  an  axis  of 
symmetry. 

VIII.  Every  straight  line  through  the  center  of  a  regular 
polygon  of  an  even  number  of  sides  cuts  the  perimeter  at  equal 
distances  from  the  center  ;  and  the  center  of  such  a  polygon  is 
a  center  of  symmetry. 

16,  Theorem, — If  the  circumference  of  a  circle  is  divided 
into  equal  arcs,  the  chords  of  those  arcs  are  the  sides  of  a  regu- 
lar polygon. 

For  the  sides  are  all  equal,  being  the  chords  of  equal  arcs ; 
and  the  angles  are  all  equal,  being  inscribed  in  equal  arcs. 


132 


PLANE  GEOMETRY. 


[Chap.  YII. 


lY.  Theorem. — If  a  circumference  is  divided  into  equal 
arcs,  and  lines  tangent  at  the  several  points  of  division  are  pro- 
duced until  they  meet,  these  tangents  are  the  sides  of  a  regular 
polygon. 

Let  A,  B,  C,  etc.,  be  points  of  division,  and  F,  D,  and  E 
points  where   the  tangents  meet. 
Join  6^^,  ^^,  and^a 

The  triangles  GAF,  ABB, 
and  BCE  have  the  sides  GA, 
AB,  and  BC  equal,  as  they  are 
chords  of  equal  arcs ;  and  the 
angles  at  G,  A,  B,  and  G  equal, 
for  each  is  formed  by  a  tangent 
and  chord  which  intercept  equal 
arcs.  Therefore,  these  triangles 
are  all  isosceles,  and  all  equal ; 
and  the  angles  F,  D,  and  E  are 
equal.     Also,  FB  and  BE,   be-  , 

ing  doubles  of  equals,  are  equal.  In  the  same  manner,  it  is 
proved  that  all  the  angles  of  the  polygon  FBE  are  equal,  and 
that  all  its  sides  are  equal.  That  is,  it  is  a  regular  polygon, 
circumscribed  about  the  circle. 


Regular  Polygons  Similar. 

18.  Theorem., — Regular  polygons  of  the  same  number  of 
sides  are  similar. 

For,  making  all  the  radii  in  the  polygons  compared,  each  is 
seen  to  be  composed  of  as  many  triangles  as  it  has  sides  (15,  iv). 
All  these  triangles  are  similar,  since  their  angles  are  respectively 
equal  ;  and  they  are  similarly  arranged  in  the  polygons.  There- 
fore, the  polygons  are  similar  (10). 

Corollaries. — I.  The  areas  of  two  regular  polygons  of  the 
same  number  of  sides  are  to  each  other  as  the  squares  of  their 
homologous  lines. 

II.  In  regular  polygons  of  the  same  number  of  sides,  any  two 
certain  lines  have  a  constant  ratio  (12). 


Art.  18.] 


POLYGONS. 


133 


For  instance,  the  ratio  of 
the  diagonal  to  the  side  of  a 
square  is  always  as  the  square 
root  of  2  is  to  1,  since  the  *\ 
square  on  the  diagonal  is  equiv- 
alent to  the  sum  of  the  squares 
on  two  sides  (13,  c). 

Similarly,  the  ratios  between 
the  side  and  apothem,  between  the  side  and  radius,  and  between 
the  radius  and  apothem  of  a  regular  polygon  of  a  certain  num- 
ber of  sides,  are  constant. 

III.  If  two  similar  regular  polygons  of  an  even  number  of 
sides  have  their  sides  respectively  parallel  when  taken  in  the 
same  order,  they  are  also  respectively  parallel  when  taken  in  re- 
verse order.  Therefore  two  such  polygons  so  placed  have  two 
centers  of  similarity,  one  external  and  one  internal  (9,  c)  ;  but, 
if  the  polygons  are  also  placed  concentrically,  the  two  centers  of 
similarity  coincide  at  the  center  of  the  polygons. 


19.  Exercises.— \.  First  in  right  angles,  and  then  in  degrees,  ex- 
press the  value  of  an  angle  of  each  regular  polygon,  from  three  sides  up 
to  ten. 

2.  First  in  right  angles,  and  then  in  degrees,  express  the  value  of 
an  angle  at  the  center,  subtended  by  one  side  of  each  of  the  same 
polygons. 

3.  The  exterior  angle  of  a  regular  polygon  being  one  third  of  a  right 
angle,  find  the  number  of  sides  of  the  polygon. 

4.  A  plane  may  be  so  covered  by  equilateral  triangles  as  to  leave  no 
intervening  surface. 


5.  A  plane  may  be  covered  in  the  same  way  by  equal  squares. 


134  PLANE  GEOMETRY.  [Chap.  VII. 

6.  A  plane  may  be  covered  in  the  same  way  by 
regular  hexagons. 

7.  A  plane  can  not  be  covered  in  the  same  way 
by  equal  regular  polygons  of  any  other  number  of 
sides. 

8.  Find  the  ratios  between  the  side,  the  radius, 
and  the  apothem,   of  a  square.     [The  apothem, 

radius,  and  half  side  of  every  regular  polygon  form  a  right-angled  tri- 
angle. If  we  know  any  one  of  the  ratios  from  the  nature  of  the  polygon, 
the  Pythagorean  Theorem  enables  us  to  find  the  others.] 

9.  Find  the  ratios  between  the  side,  the  radius,  and  the  apothem,  of 
a  regular  hexagon.  [What  sort  of  a  triangle  is  made  by  a  side  and  two 
radii?] 

10.  The  area  of  an  inscribed  regular  hexagon  is  three  fourths  that  of 
a  regular  hexagon  circumscribed  about  the  same  circle. 


Maxima  and  Minima. 

30.  A  Maximum  magnitude  is  the  greatest  of  its 
kind. 

A  Miuimum  magnitude  is  the  least  of  its  kind. 

For  example,  a  diameter  is  the  maximum  chord  in  a  given 
circle  ;  and  a  perpendicular  is  the  minimum  line  from  a  given 
point  to  a  given  straight  line. 

Isoperimetrical  Figures  are  inclosed  surfaces  whose 
perimeters  have  the  same  extent. 

Theorem. — The  shortest  line  that  extends  from  one  point  to 
another,  through  some  point  of  a  given  straight  line,  mahes 
equal  angles  with  that  line. 

Suppose  first  that  the  points  are  on  opposite  sides  of  the  line. 
Then  the  shortest  line  joining  them  is  a  straight  line  cutting  the 
given  line  and  making  equal  angles. 

If  the  points  are  on  the  same  side  of  the  straight  line, 
the  shortest  line  from  one  to  the  other  through  a  point  of 
that  line  must  be  composed  of  two  straight  lines.  Let 
CD  be  the  line  and  A  and  B  the  points,  and  AEB  the 
shortest  line  that  can  be  made  from  A  to  B  through  any 
point  of  CD,  It  is  to  be  proved  that  AEG  and  BUB  are 
equal  angles. 


Art.  20.] 


POLYGONS. 


135 


Make  ^J9^  perpendicular  to  CD,  and  produce  it  to  F,  making 
HF  equal  to  AH. 

Every  point  of  the  line  CD  is  equally  distant  from  A  and 
F.  Therefore,  every  line  joining  B  to  F  through  some  point  of 
CD  is  equal  to  a  line  joining 
jB  to  ^  through  the  same 
point.  Thus  BGF  is  equal 
to  B  GA,  and  BFF  is  equal 
to  BFA.  Since  BFA  is  the 
shortest  line  from  B  to  A 
through  any  point  of  CD, 
BFF  is  the  shortest  line  from 
B  o  F  and  it  is  therefore  a 
straight  line. 

Then  the  angles  FFH  and  BFD,  being  vertical,  are  equal ; 
but  FFjB:  and  AFBT  are  equal  (III,  10,  iv).  Therefore  AFBT 
and  BFD  are  equal. 


21.  Theorem. —  Of  all  equivalent  triangles  of  a  given  hose, 
the  one  having  the  least  perimeter  is  isosceles. 

The  equivalent  triangles  having  the  same  base,  AE,  have 
also  the  same  altitude,  and  their  vertices  are  in  the  same  line 
parallel  to  the  base. 

The  shortest  line  that  can 
be  made  from  ^  to  ^  through 
some  point  of  DB  constitutes 
the  other  two  sides  of  the  tri- 
angle of  least  perimeter.  This 
shortest  line  is  the  one  making 

equal  angles  with  DB,  that  is,  making  A  CD  and  FCB  equal. 
The  angle  A  CD  is  equal  to  its  alternate  A,  and  the  angle  FCB 
to  its  alternate  F.  Therefore,  the  angles  at  the  base  are  equal, 
and  the  triangle  is  isosceles. 


22.  Theorem. —  Of  all  equivalent  polygons  of  the  same 
number  of  sides,  the  one  having  the  least  perimeter  is  regu- 
lar. 


136 


PLANE  GEOMETRY. 


[Chap.  VII. 


Of  several  equivalent  polygons,  suppose  AB  and  ^  C  to  be 
two  adjacent  sides  of  the  one  having  the  least  perimeter.  It  is 
to  be  proved,  first,  that  these  sides  are  equal. 

Join  AC.  Now,  if  AB 
and  BG  were  not  equal, 
there  could  be  constructed 
on  the  base  ^  (7  an  isosceles 
triangle  equivalent  to  AB  (7, 
whose  sides  would  have  less 
extent.  Then,  this  new  tri- 
angle, with  the  rest  of  the 
polygon,  would  be  equivalent 

to  the  given  polygon,  and  have  a  less  perimeter,  which  is  con- 
trary to  the  hypothesis. 

It  follows  that  AB  and  B  G  must  be  equal.  So  of  every  two 
adjacent  sides.     Therefore,  the  polygon  is  equilateral. 

It  remains  to  be  proved  that  the  polygon  has  all  its  angles 
equal. 

AB,  BG,  and  GI)  being 
consecutive  sides,  produce 
AB  and  GD  till  they  meet 
at  E.  Now  the  triangle 
BGE  is  isosceles.  For,  if 
EG,  for  example,  were  longer 
than  EB,  we  could  then  take 
EI  equal  to  jES,  .and  EF 

equal  to  EG,  and,  joining  EI,  make  the  two  triangles  EB  G  and 
EIF  equal  (V,  11). 

Then,  the  new  polygon,  having  AFID  for  part  of  its 
perimeter,  would  be  equivalent  and  isoperimetrical  to  the 
given  polygon  having  ABGD  as  part  of  its  perimeter.  But 
the  given  polygon  has,  by  hypothesis,  the  least  possible  peri- 
meter, and,  as  just  proved,  its  sides  AB,  BG,  and  GD  are 
equal. 

If  the  new  polygon  has  the  same  area  and  perimeter,  its  sides 
also  must  be  equal ;  that  is,  AF,  FI,  and  ID.  But  this  is  ab- 
surd, for  AF  is  less  than  AB,  and  ID  is  greater  than  GD. 
Therefore,  the  supposition  that  EG  is  greater  than  EB,  which 


Art.  22.]  POLYGONS,  137 

supposition  led  to  this  conclusion,  is  false.  Hence,  EB  and  JEC 
are  equal. 

Therefore,  the  angles  EBQ  and  EGB  are  equal,  and  their 
supplements  ABC  and  BCD  are  equal.  Thus,  it  is  shown  that 
consecutive  angles  are  equal. 

It  being  proved  that  the  polygon  has  its  sides  equal  and  its 
angles  equal,  it  is  regular. 


33.  Theorem. —  Of  all  isoperimetrical polygons  of  the  same 
number  of  sides,  that  which  is  regular  has  the  greatest  area. 

Compare  two  isoperimetrical  polygons  of  the  same  number  of 
sides,  one  of  the  two  being  regular.  Designate  the  regular  poly- 
gon as  By  the  other  as  I.  There  may  be  a  third  polygon 
similar  to  I  and  equivalent  to  B  (II,  3). 

By  the  last  theorem,  the  perimeter  of  this  third  polygon 
is  greater  than  that  of  B,  therefore  greater  than  the  perimeter 
of  I.  As  the  third  polygon  is  similar  to  I  and  has  a  greater 
perimeter,  it  must  have  a  greater  area  (13,  c).  Therefore  its 
equivalent  B  has  a  greater  area  than  I, 

24,  Theorem. —  Of  regular  equivalent  polygons,  that  which 
has  the  greatest  number  of  sides  has  the  least  perimeter. 

It  will  be  sufficient  to  demonstrate  the  principle,  when  one  of 
the  equivalent  polygons  has  one  side  more  than  the  other. 

In  the  polygon  having  the  less  number  of  sides,  join  the  ver- 


tex C  to  any  point,  as  H,  of  the  side  B  G,     Then,  on  CH  con- 
struct an  isosceles  triangle,  GKI£,  equivalent  to  CBS, 

7 


138  PLANE  GEOMETRY,  [Chap.  VII. 

Then  JIK  and  KC  are  less  than  SB  and  B  G ;  therefore, 
the  perimeter  GHKCDF  is  less  than  the  perimeter  of  its 
equivalent  polygon  GB  CDF.     But  the  perimeter  of  the  regu- 


lar polygon  AO  \s>  less  than  the  perimeter  of  its  equivalent 
irregular  polygon  of  the  same  number  of  sides,  GHKCDF, 
Therefore,  it  is  less  than  the  perimeter  of  GB  CDF. 

25.  Theorem, —  Of  several  regular  isoperimetrical polygons 
the  greatest  is  that  which  has  the  greatest  number  of  sides. 

Designate  two  regular  isoperimetrical  polygons,  one  having 
m  sides,  as  M:  and  one  having  more  than  m  sides,  as  N',  There 
may  be  a  third  polygon  similar  to  M  and  equivalent  to  JST, 

By  the  last  theorem,  the  perimeter  of  this  third  polygon  is 
greater  than  of  N",  or  its  equal,  the  perimeter  of  3f,  As  this 
third  polygon  is  similar  to  M  and  has  a  greater  perimeter,  it 
must  have  a  greater  area  (13,  c).  Therefore  its  equivalent  iV 
has  a  greater  area  than  M. 


26.  Application. — When  a  ray  of  light  is  reflected,  the  incident 
and  reflected  parts  of  the  ray  make  equal  angles  with  the  surface.  That 
is,  light,  when  reflected,  adheres  to  the  law  that  requires  it  to  take  the 
shortest  path. 

27.  Exercises. — 1.  Three  houses  are  built  with  walls  of  the  same 
aggregate  length ;  the  first  in  the  shape  of  a  square,  the  second  of  a 
rectangle,  and  the  third  of  a  regular  octagon.  Which  has  the  greatest 
amount  of  room,  and  which  the  least? 


Art.  27.]  POLYGONS,  139 

2.  Of  all  equivalent  parallelograms  having  equal  bases,  what  one  has 
the  minimum  perimeter? 

3.  Of  all  triangles  having  two  sides  respectively  equal  to  two  given 
lines,  the  greatest  is  that  where  the  angle  included  between  the  given 
sides  is  a  right  angle. 

4.  In  order  to  cover  a  pavement  with  equal  blocks,  in  the  shape  of 
regular  polygons  of  a  given  area,  of  what  shape  must  they  be  that  the 
entire  extent  of  the  lines  between  the  blocks  shall  be  a  minimum  ? 

S8.  Scholium. — Certain  principles  which  had  been  demon- 
strated as  to  equality  and  similarity  of  triangles  and  as  to  cen- 
ters of  similarity  have  been  generalized  in  this  chapter.  That 
is,  they  are  proved  to  be  applicable  to  all  polygons.  The  knowl- 
edge of  the  regular  polygon  has  been  added,  and  some  princi- 
ples of  maxima  and  minima. 

29.  Miscellaneous  Exercises. — 1.  Show  that  the  side,  the 
radius,   and  the  apothem,   of    a  regular  triangle  have   the    ratios    of 

2V^3;  2;  1. 

2.  Show  that  the  corresponding  ratios  in  a  regular  octagon  are  as  2 ; 

V4+  2  V2;  (1  +  /2). 

3.  If  two  diagonals  of  a  regular  pentagon  cut  each  other,  both  are 
divided  in  extreme  and  mean  ratio.  [Circumscribe  a  circle  about  the 
polygon.     Compare  V,  44,  xi.] 

4.  All  the  diagonals  being  formed  in  a  regular  pentagon,  the  figure 
inclosed  by  them  is  a  regular  pentagon. 

6.  Show  that  the  side,  the  radius,  and  the  apothem,  of  a  regular  pen- 
tagon have  the  ratios  of  i  i^lO  —  2  V"5 ;  1 ;  i  (1  +  V^). 

6.  If  a  regular  pentagon,  hexagon,  and  decagon  are  inscribed  in  a 
circle,  a  triangle  having  its  sides  respectively  equal  to  the  sides  of  these 
three  polygons  is  right-angled.     [YI,  23,  c] 

7.  If,  from  any  point  within  a  given  regular  polygon,  perpendiculars 
fall  on  all  the  sides,  the  sura  of  these  perpendiculars  is  a  constant  quan- 
tity. [Make  a  line  from  the  point  named  to  every  vertex  of  the  poly- 
gon.] 


140  PLANE  GEOMETRY.  [Chap.  VU. 

Problems  in  Drawing. 

30.  Problems. — I.   To  draw  a  polygon  equal  to  a  given  polygon. 

II. — To  draw  a  polygon  when  all  its  sides  and  all  the  diagonals  from 
one  vertex  are  given  in  their  order. 

These  depend  on  Y,  44,  i. 

III. — To  draw  a  polygon  when  the  sides  and  angles  are  given  in  their 
order. 

It  is  enough  for  this  problem  if  all  the 
angles  except  three  are  given.  Suppose 
first  that  the  angles  not  given  are  consecu- 
tive, as  at  D,  Bj  and  C.  Draw  the  trian- 
gles «,  c,  i,  and  0.  Having  DC,  complete 
the  polygon  by  drawing  the  triangle  DBG 
from  its  three  known  sides.  Suppose  the 
angles  not  given  are  2>,  G,  and  F.  Then 
draw  the  triangles  a,  e,  and  i,  and  separ- 
ately, the  triangle  u.  Having  the  three  sides  of  the  triangle  o,  it  may 
be  drawn,  and  the  polygon  completed. 

The  problem,  to  iDscribe  a  regular  polygon  in  a  circle  by- 
means  of  straight  lines  and  arcs  of  circles,  can  be  solved  in  only 
a  limited  number  of  cases.  The  solution  depends  upon  the 
divisiop  of  the  circumference  into  equal  arcs  ;  and  this  depends 
upon  the  division  of  the  sum  of  four  right  angles  into  equal 
parts. 

ly. — To  inscribe  a  square  in  a  given  circle. 

Draw  two  diameters  perpendicular  to  each  other.  Join  their  extrem- 
ities by  chords.     These  chords  form  an  inscribed  square. 

For  the  angles  at  the  center  are  equal  by  construction.  Therefore, 
the  subtending  arcs  are  equal,  and  the  chords  of  those  arcs  are  the  sides 
of  a  regular  polygon. 

V. — To  inscribe  a  regular  hexagon  in  a  circle. 

Suppose  the  figure  completed;   then 

drawing  two  successive    radii,   we  have  ^^ ^ 

the  triangle  ABG.  /f  Y\ 

The  angle  G  is  60°  (15,  iii).    Hence  n  \\ 

the  sum  of  the  angles  at  A  and  B  is  120°,  //  \l 

^  V  C t  A 

and,  as  they  are  equal,  each  is  60°.    There-  K  \  A 

fore,  the  triangle  is  equilateral,  and  one  \\  \       /I 

side  of  a  regular  hexagon  is  equal  to  its  \\ \l^ 

radius.  ^- P^-ff 


Art.  30.]  POLYGONS,  141 

The  solutiou  of  the  problem  is— apply  the  radius  to  the  circumfer- 
ence six  times  as  a  chord. 

YI. — To  inscribe  a  regular  decagon  in  a  circle. 

Divide  the  radius  CA  in  extreme  and 
mean  ratio  at  the  point  B  (V,  44,  x), 
making  BG  the  greater  part.  Draw  the 
chord  AD  equal  to  BG.  This  chord  is 
one  side  of  the  inscribed  decagon. 

For  the  angle  G  is  one  fifth  of  a 
straight  angle  (Y,  44,  xi).  Hence,  the 
arc  AD  is  one  tenth  of  the  circumfer- 
ence. 

Corollary, — Many  other  regular  poly- 
gons may  be  drawn  by  means  of  the  above 
three. 

To  inscribe  an  equilateral  triangle,  join  the  alternate  vertices  of  the 
hexagon. 

To  inscribe  a  regular  pentagon,  join  the  alternate  vertices  of  the 
decagon. 

To  inscribe  a  regular  pentedecagon,  subtract  the  arc  subtended  by  the 
side  of  the  decagon,  36°,  from  that  subtended  by  the  side  of  the  hexa- 
gon, 60°.  The  remainder,  24°,  is  one  fifteenth  of  the  circumfer- 
ence. 

Having  an  inscribed  regular  polygon,  to  inscribe  one  of  double  the 
number  of  sides,  bisect  each  arc  subtended  by  a  side  of  the  given  polygon, 
and  draw  the  chords  of  all  these  half  arcs. 

YH. — To  circumscribe  a  regular  polygon  about  a  circle. 

Divide  the  circumference  into  the  requisite  number  of  equal  arcs  by 
the  above  methods  for  inscribing  regular  polygons.  Through  the  points 
of  division  draw  tangents.  These,  produced  til]  they  meet,  form  the  re- 
quired polygon  (17). 

Scholium. — It  has  been  demonstrated  by  Gauss  that  any 
regular  polygon  can  be  drawn,  using  only  straight  lines  and  arcs 
of  circles,  if  the  number  of  sides  is  prime  and  is  included  in 
some  value  of  the  expression  2"  -j-  1,  the  exponent  n  being  any 
whole  number.  This  includes  the  following  numbers  :  3 ;  5  ; 
17  ;  257  ;  and  65537.  No  other  numbers  included  in  this  for- 
mula are  known  to  be  prime.  Any  other  such  prime  number 
must  exceed  the  sixty-fourth  power  of  2, 


143  PLANE  GEOMETRY.  [Chap.  VII. 

31.  Exercises  in  Drawing. — 1.  To  draw  a  quadrilateral  when 
the  four  sides  and  one  angle  are  given. 

2.  To  draw  a  parallelogram  when  two  adjacent  sides  and  an  angle 
are  given. 

3.  To  draw  a  parallelogram,  having  the  diagonals  and  one  side  given. 

[VI,  6.] 

4.  To  bisect  any  quadrilateral  by  a  line  from  a  given  vertex,  that  is, 
to  divide  it  into  two  equivalent  parts. 

5.  To  draw  the  mimimum  tangent  from  a  given  straight  line  to  a 
given  circumference.     [Ill,  10.] 

6.  To  draw  a  triangle  similar  to  a  given  triangle,  but  with  double  the 
area.    [13,  o.] 

7.  To  construct  a  regular  octagon  of  a  given  side. 

8.  Given  a  regular  inscribed  polygon,  to  circumscribe  a  similar  poly- 
gon whose  sides  are  parallel  to  the  former. 

9.  To  inscribe  a  square  in  a  given  segment  of  a  circle. 


CHAPTER  VIII. 
CIRCLES. 

Article  1. — The  properties  of  circumferences  and  of 
straight  lines  in  connection  with  them  have  been  developed  in 
Chapter  IV. 

A  Segment  of  a  circle  is  the 
part  cut  off  by  a  secant  or  chord  ; 
asAJBC.oT  GDE. 

A  Sector  of  a  circle  is  the 
part  between  two  radii  and  the 
arc  intercepted  by  them ;  as 
GRL 

The  liimit  of  Polygons. 

3.  Theorem. — If  a  polygon  inscribed  in  a  circle  is  made  to 
vary  by  bisecting  every  arc  subtended  by  a  side  and  Joining  these 
midpoints  to  the  ends  of  the  sides  so  as  to  double  the  number  of 
sides  of  the  polygon  at  every  step,  the  circle  is  the  limit  of  this 
variable  polygon. 

Also,  if  a  circumscribed  polygon  is  made  to  vary  by  bisect- 
ing every  arc  between  the  points  of  tangency  and  making  tan- 
gents at  these  midpoints  so  as  to  double  the  number  of  sides  at 
every  step,  the  circle  is  the  limit  of  this  variable  circumscrihed 
polygon. 

That  is,  the  variable  polygon  can  approximate  the  circle 
within  less  than  any  assignable  difference,  but  can  never  become 
equal  to  or  pass  it. 


144 


PLANE  GEOMETRY. 


[Chap.  Via 


Suppose  the  chord  AB  a  Bide  of  an  inscribed  polygon,  and 
CD  and  CJE  sides  of  a  circumscribed  polygon,  tangent  at  A 
and  B. 

Make  the  tangent  FG  par- 
allel to  AB,  and  join  AH  and 
BH,  By  a  similar  construction 
at  every  arc  subtended  by  a  side 
of  the  inscribed  polygon,  we 
have  a  second  inscribed  polygon, 
having  for  two  of  its  sides  the 
chords  AH  and  HB^  and  a  new 
circumscribed  polygon  having  for 
part  of  its  perimeter  the  tangents 
AF,  FG,  and  GB, 

The  difference  between  the 
first  inscribed  and  the  first  circumscribed  polygon  is  the  sum  of 
the  triangle  ABC  and  the  other  similarly  situated  triangles. 
The  difference  between  the  second  inscribed  and  the  second  cir- 
cumscribed polygon  is  the  sum  of  the  triangles  AFH,  HB  G, 
and  the  others  similarly  situated. 

The  triangle  AHB,  having  the  same  altitude  as  the  trape- 
zoid AFGB  and  its  base  the  greater  base  of  the  trapezoid, 
has  more  than  half  the  area  of  the  trapezoid.  Hence,  the 
area  of  the  triangles  AFH  and  HGB  is  less  than  half  the 
area  of  the  trapezoid,  and  much  less  than  half  of  the  triangle 
ABC. 

As  the  same  reasoning  applies  to  the  construction  upon  every 
side,  the  second  inscribed  and  the  second  circumscribed  polygon 
approximate  each  other  by  more  than  half  the  previous  differ- 
ence of  extent. 

Repeating  this  process,  the  inscribed  and  circumscribed  poly- 
gons are  variable  figures  which  approximate  each  other  in  such 
a  way  that  less  than  half  of  the  difference  of  their  areas  remains 
at*every  step.  It  can  in  this  manner  become  less  than  any 
designated  extent  of  surface.  But  at  every  step  the  circle  con- 
tains the  inscribed  polygon  and  is  therefore  greater  than  it ; 
and  is  contained  by  the  circumscribed  polygon  and  is  therefore 
less  than  it. 


Art.  2.]  CIRCLES,  145 

Neither  perimeter  can  ever  pass  the  curve,  nor  coincide  with 
it,  for,  however  short  the  sides  may  be,  they  are  straight  lines. 
The  circle  is  therefore  the  limit  of  both  the  variable  poly- 
gons. 

The  variable  polygon  may  be  regular  or  not,  and  the  curve 
may  be  that  of  the  circle  or  any  other. 

Corollaries. — I.  The  circumference  is  the  limit  of  the 
perimeters  of  the  variable  inscribed  and  circumscribed  poly- 
gons. 

11.  The  superficial  ratio  of  similar  plane  figures,  whether 
curvilinear  or  rectilinear,  is  the  square  of  their  linear 
ratio. 

3.  Theorem. — A  curve  is  shorter  than  any  line  thai  Joins 
its  ends,  and  toioard  which  it  is  convex. 

For   a  broken  line  having  its 
vertices  in  JBD  G  may  be  made  to  jP 

approximate  BDG  within  less 
than  any  assignable  difference. 
But  such  broken  line  is  less  than 
BFC  (III,  3).     Therefore,  BDG  " 

must  be  less  than  BFG. 

Corollary. — The  circumference  of  a  circle  is  shorter  than  the 
perimeter  of  a  circumscribed  polygon. 

That  the  circumference  is  longer  than  the  perimeter  of  any 
inscribed  polygon,  follows  immediately  from  the  Axiom  of  Dis- 
tance. 

4.  Theorem. — A  circle  has  a  less  perimeter  than  any  equiva- 
lent polygon. 

For,  of  equivalent  polygons,  that  has  the  least  perimeter 
which  is  regular  (VII,  22),  and  has  the  greatest  number  of  sides 
(VII,  24). 

Corollary. — A  circle  has  a  greater  area  than  any  isoperi- 
metrical  figure. 


146  PLANE  GEOMETRY.  [Chap.  VIII. 

Similarity  of  Circular  Figures. 

5.   Theorem. —  Circles  are  similar  figures. 

Whatever  lines  are  made  in  one  circle,  homologous  lines 
making  equal  angles  may  be  made  in  another  (IV,  20  to  26). 

Corollaries. — I.  Two  arcs  are  similar  when  they  subtend 
equal  angles  at  the  center. 

II.  Sectors  are  similar  when  their  arcs  are  similar.  Segments 
likewise. 

III.  Similar  arcs  have  the  same  amount  of  curvature ;  and 
conversely. 

6«  Theorem, — The  ratio  of  the  circumference  to  the  diam- 
eter is  a  constant  quantity. 

For  the  perimeters  of  similar  polygons  have  the  same  ratio 
to  homologous  lines.  Then,  the  circumferences,  which  are  the 
limits  of  the  perimeters,  must  have  the  same  ratio  to  their  re- 
spective diameters. 

This  constant  ratio  is  designated  by  the  Greek  letter  tt,  the 
initial  of  perimeter.  Then,  if  the  radius  is  H,  the  circumference 
is  27ri2. 

Corollaries. — ^I.  Any  homologous  lines  in  similar  circular 
figures  (that  is,  circles  or  parts  of  circles)  are  proportional. 

II.  The  curvatures  of  two  equivalent  arcs  of  different  radii 
are  inversely  as  the  lengths  of  their  radii. 

III.  Of  two  arcs  having  a  common  chord,  that  having  less 
curvature  has  less  extent. 

lY.  If  two  minor  arcs  have  a  common  chord,  the  shorter  arc 
is  the  one  that  has  the  longer  radius  ;  and  conversely. 

V.  If  two  major  arcs  have  a  common  chord,  the  shorter  arc 
has  the  shorter  radius  ;  and  conversely. 

7.  Theorem.. — Any  two  circles  in  one  plane  have  an  inter- 
nal and  an  external  center  of  similarity ,  viz.,  those  points  in  the 
line  joining  the  centers,  where  that  line  is  divided  in  the  ratio  of 
the  radii. 

Let  B  and  C  be  the  centers  of  two  circles,  and  F  and  G 


Art.  7.]  CIRCLES.  14T 

the  points  where  the  central  line  is  divided  internally  and  exter- 
nally in  the  ratio  of  the  two  radii. 


Let  BH  and  CE  be  any  two  parallel  radii  lying  in  the  same 
direction  from  B  and  O.     Then 

aB:GC  =  BH'.CE, 

Let  BH  and  CD  be  any  two  parallel  radii,  in  opposite  di- 
rections from  B  and  C. 
Then 

FB\FG-BH\  CD, 

Therefore,  the  points  6r,  E^  and  H  are  in  one  straight  line  ; 
also,  the  points  Z>,  F,  and  11. 
Compare  VII,  18,  iii. 


Rectification  of  the  Circumference. 

8.  The  RectifLcation  of  a  curve  consists  in  finding  a 
straight  line  of  the  same  extent.  It  has  been  demonstrated  that 
this  cannot  be  done  for  the  circumference  of  a  circle  by  the 
methods  allowed  in  problems  in  drawing  ;  but  the  length  of  the 
curve  may  be  calculated  to  any  required  degree  of  approxima- 
tion. 

The  number  tt  is  less  than  4  and  greater  than  3.  For,  if  the 
diameter  is  1,  the  perimeter  of  the  circumscribed  square  is  4, 
but  this  is  greater  than  the  circumference  ;  and  the  perimeter  of 


148 


PLANE  GEOMETRY. 


[Chap.  VIU. 


the  inscribed  regular  hexagon  is  3,  but  this  is  less  than  the  cir- 
cumference. To  calculate  this  number  more  accurately,  use  the 
following 

Theorem. —  When  one  of  two  isoperimetrical  polygons  has 
half  as  many  sides  as  the  other, 

1.  The  apothem  of  the  second  is  half  the  sum  of  the  radius 
and  apothem  of  the  first ;  and 

2.  The  radius  of  the  second  is  a  inean  proportional  between 
the  radius  of  the  first  and  the  apothem  of  the  second. 

Let  ^0  be  the  apothem  and  BO 
the  radius  of  the  first  polygon.  Then, 
making  the  right-angled  triangle  BA  O, 
AB  is  half  the  side  of  the  polygon,  O 
is  the  center,  and  BFG,  described  with 
OB  as  a  radius,  is  an  arc  of  the  circum- 
scribed circle. 

Produce  AO  to  the  circumference 
at  (7,  join  BC,  and  make  01)  perpen- 
dicular to  B  Cy  and  I)£J  perpendicular 

to^a 

The  line  OB  being  half  the  line  CB  (IV,  11,  i),  JEJB  is  half 
of  AB  (Y,  20,  ii).  Therefore,  JEJB  is  half  of  the  side  of  the 
second  polygon,  which  has  double  the  number  of  sides  and  same 
perimeter  as  the  first.  Since  the  angle  ECD  is  half  of  the  angle 
A  OB  (lY,  22,  i),  DG\^  the  radius  and  EC  the  apothem  of  the 
second  polygon. 

Now,  EG  is  half  of  A  G,  that  is,  half  of  the  sum  of  the 
apothem  and  radius  of  the  first  polygon.  In  the  right-angled 
triangle  0J9C  (Y,  29) 

CO'.GD=GB:  GE. 

That  is,  the  radius  of  the  second  polygon  is  a  mean  proportional 
between  the  radius  of  the  first  polygon  and  the  apothem  of  the 
second. 

Representing  the  apothem  of  the  first  polygon  by  a,  the 
radius  by  r,  and  the  apothem  of  the  polygon  of  double  the  num- 
ber of  sides  by  x,  and  its  radius  by  y, 

x-=.  \{a-\-r),  and  y  =  V  ^^' 


Art.  9.]  CIRCLES.  149 

9,  Problem, — To  find  the  approximate  value  of  the  ratio  of 
the  circumference  to  the  diameter  of  a  circle. 

In  any  regular  polygon,  the  apothem,  the  radius,  and  half  the 
side  form  a  right-angled  triangle.     Hence, 


r^  =  a^  +  (|.s)3...  a=  ^^s  —  i^s  z=  ^  \/ 4.r^ —s^. 

Suppose  a  regular  hexagon,  whose  perimeter  is  unity.  Then 
its  side  is  ■}-  or  .167  nearly,  and  its  radius  is  the  same.     The 

apothem  h  ^  V  f^  —  "sV  =  1*2-  '^  ^>  ^^  -1^^  +• 

Then,  by  the  formula,  x  ^  \  {a  -\-  r),  find  the  apothem  of  the 
regular  dodecagon,  whose  perimeter  is  unity.  It  is  .156  nearly. 
The  radius  of  the  same,  by  the  formula,  y  =  V  a;r,  is  .161  nearly. 
Proceeding  in  this  way,  find  the  apothem  and  radius  of  the  iso- 
perimetrical  polygon  of  24  sides,  and  so  on. 


Number  op 

Sides. 

Apothem. 

Radius. 

6 

.144  + 

.167  — 

12 

.156  — 

.161  — 

24 

.158  + 

.160  — 

48 

.159  — 

.159  + 

Since  the  perimeter  in  the  above  calculation  is  always  unity, 
these  figures  indicate  the  ratios  of  the  apothems  and  radii  of  the 
polygons  to  their  respective  perimeters.  These  are  the  same  as 
the  ratios  of  the  radius  of  a  circle  to  the  perimeters  of  the  cir- 
cumscribed and  inscribed  polygons.  Since  the  circumference  is 
less  than  the  former  and  greater  than  the  latter,  the  ratio  of  the 
radius  to  the  circumference  must  be  between  the  values  of  any 
pair  of  the  above  ratios.  It  is  shown,  at  the  first  step,  that  it 
must  be  greater  than  .144  and  less  than  .167. 

As  the  calculation  proceeds,  the  ratios  approximate.  As  far 
as  they  agree,  the  ratio  of  the  radius  to  the  circumference  of  a 
circle  is  ascertained.  Hence  we  know  that  .159  is  this  ratio 
within  one  thousandth.  By  calculating  to  a  greater  number  of 
places,  and  continuing  the  table,  the  ratio  may  be  calculated  to 
a  greater  degree  of  accuracy.  The  ratio  of  the  diameter  to  the 
circumference  must  be  twice  .159  =  .318.     Therefore, 

TT  =  1  -f-  .318  =  3.14  +. 


150  PLANE  GEOMETRY,  [Chap.  VIII. 

10.  Exercises. — 1.  When  the  Tyrian  Princess  stretched  the 
thongs  cut  from  the  hide  of  a  bull  around  the  site  of  Carthage,  what 
course  should  she  have  pursued  in  order  to  include  the  greatest  extent  of 
territory  ? 

2.  If  two  circles  touch  each  other,  any  line  through  the  point  of  con- 
tact cuts  off  similar  segments. 

3.  What  is  the  locus  of  the  midpoints  of  the  lines  extending  from  a 
given  point  to  a  given  circumference  ? 

4.  If  two  circles  touch  each  other,  any  two  straight  lines  extending 
through  the  point  of  contact  are  cut  proportionally  by  the  circumfer- 
ences. 

5.  If  two  circles  are  tangent  to  each  other  externally  and  to  a  straight 
line  on  the  same  side  of  them,  that  part  of  the  common  tangent  which  is 
between  the  points  of  contact  is  a  mean  proportional  between  the  two 
diameters.  [At  the  point  of  contact  of  the  two  circles,  erect  a  perpen- 
dicular to  the  central  line ;  from  the  point  where  this  meets  the  tangent, 
make  a  line  to  the  center  of  each  circle.] 

6.  Two  wheels,  whose  diameters  are  twelve  and  eighteen  inches,  are 
connected  by  a  belt,  so  that  the  rotation  of  one  causes  that  of  the  other. 
The  smaller  makes  twenty-four  rotations  in  a  minute ;  what  is  the  velocity 
of  the  larger  wheel  ? 

7.  Two  wheels,  whose  diameters  are  twelve  and  eighteen  inches,  are 
fixed  on  the  same  axle,  so  that  they  turn  together.  A  point  on  the  rim 
of  the  smaller  moves  at  the  rate  of  six  feet  per  second,  what  is  the  ve- 
locity of  a  point  on  the  rim  of  the  larger  wheel  ? 

8.  If  the  radius  of  a  car-wheel  is  thirteen  inches,  how  many  rotations 
does  it  make  in  traveling  one  mile? 

9.  If  the  equatorial  diameter  of  the  earth  is  7926  miles,  what  is  the 
length  of  one  degree  of  longitude  on  the  equator? 


Quadrature  of  tlie  Circle. 

11.  The  Quadrature  or  Squaring  of  a  Circle  consists 
in  finding  an  equivalent  rectilinear  figure. 

Theorem. — The  area  of  a  circle  is  equal  to  half  the  product 
of  its  circumference  hy  its  radius. 

First  consider  the  area  of  any  polygon,  circumscribed  about 
a  circle,  as  ABDEF. 

From  the  center  of  the  circle,  let  straight  lines  extend  to  the 
vertices  of  the  polygon,  also  to  the  points  of  tangency. 

The  lines  extending  to  the  points  of  tangency  are  radii,  and 


Art.  11.]  CIRCLES.  151 

therefore  perpendicular  to  the  sides  of  the  polygon,  which  are 
tangents  of  the  circle.  The  polygon  is  divided  by  the  lines  ex- 
tending to  the  vertices  into  as  many  triangles  as  it  has  sides, 
ACB,  BCD,  etc.  Regarding  the  sides  of  the  polygon,  AB^ 
BBj  etc.,  as  the  bases  of  these  several  triangles,  they  have  equal 


altitudes.  Now,  the  area  of  each  triangle  is  equal  to  half  the 
product  of  its  base  by  the  common  altitude.  But  the  area  of 
the  polygon  is  the  sum  of  the  areas  of  the  triangles,  and  the 
perimeter  of  the  polygon  is  the  sum  of  their  bases.  It  follows 
that  the  area  of  the  polygon  is  equal  to  half  the  product  of  the 
perimeter  by  the  radius. 

As  this  is  true  of  all  circumscribed  polygons,  it  is  true  of 
the  circle,  which  is  the  limit  of  a  variable  circumscribed  poly- 
gon. 

Corollaries. — I.  The  area  of  a  regular  polygon  is  measured 
by  half  the  product  of  its  perimeter  by  its  apothem. 

II.  Let  r  represent  the  radius.  Then  the  circumference  is 
2  TT  r,  and  half  the  product  of  this  by  r  is  tt  r^  ;  that  is,  the  area 
is  equal  to  the  square  of  the  radius  multiplied  by  the  ratio  of  the 
circumference  to  the  diameter. 

III.  The  area  of  a  circle  is  to  the  square  described  on  the 
radius  as  the  circumference  is  to  the  diameter.  This  shows  that 
the  problems  of  rectification  and  of  quadrature  are  in  effect 
the  same. 

IV.  The  areas  of  two  circles  are  to  each  other  as  the  squares 
of  their  radii ;  or,  as  the  squares  of  their  diameters. 


152  PLANE  GEOMETRY.  [Chap.  VIII. 

V.  The  area  of  a  sector  is  measured  by  half  the  product  of 
its  arc  by  its  radius.  For  the  sector  is  to  the  circle  as  its  arc  is 
to  the  circumference. 


13.  Exercises. — 1.  What  is  the  length  of  the  radius  when  the  arc 
of  60°  is  10  feet? 

2.  What  is  the  value,  in  degrees,  of  the  angle  at  the  center,  whose  arc 
has  the  same  length  as  the  radius  ? 

3.  What  is  the  area  of  the  segment  whose  arc  is  60**  and  radius  1 
foot  ?     [Subtract  the  triangle  from  the  sector.] 

4.  One  tenth  of  a  circular  field,  of  one  acre,  is  in  a  path  extending 
round  the  whole ;  required  the  width  of  the  path. 

5.  A  semicircle  described  on  the  hypotenuse  as  a  diameter  is  equiva- 
lent to  the  sum  of  the  two  semicircles  described  on  the  sides  of  a  right- 
angled  triangle. 


13,  Scholium, — It  was  shown  by  Archimedes  that  the  value 
of  TT  is  less  than  3|,  and  greater  than  S^J-.  The  number  3^  is  in 
common  use  for  mechanical  purposes. 

About  the  year  1600,  Metius  found  the  approximation  f  ff, 
which  is  true  to  six  places  of  decimals. 

By  the  calculus,  shorter  methods  have  been  discovered  for 
calculating  the  approximate  value  of  tt.  The  calculation  has 
been  extended  to  several  hundred  decimals.  The  first  thirty- 
nine  are 

3.141  592  653  589  793  238  462  643  383  279  502  884  197. 


14.  Miscellaneous  Exercises,— Tha  following  may  be  used 
while  reviewing  the  Plane  Geometry. 

1.  Take  some  principle  of  general  apphcation,  and  state  all  its  conse- 
quences that  are  found  in  the  chapter  under  review,  arranging  as  the 
first  class  those  immediately  deduced  from  the  given  principle,  then 
those  derived  from  these,  and  so  on. 

2.  Reversing  the  above,  take  some  theorem  in  the  latter  part  of  a 
chapter,  state  all  the  principles  upon  which  its  proof  immediately  de- 
pends, then  all  upon  which  these  depend,  and  so  on,  back  to  the  ele- 
ments of  the  science. 


Art.  14.]  CIRCLES.  153 

8.  If  two  straight  lines  are  not  parallel,  the  difference  between  the 
alternate  angles  formed  by  any  secant  is  constant. 

4.  If  two  opposite  sides  of  a  parallelogram  are  bisected,  straight  lines 
from  the  points  of  bisection  to  the  opposite  vertices  trisect  the  diag- 
onal. 

5.  In  any  triangle  ABG^  if  BE  and  OF  are  perpendiculars  to  any 
line  through  A^  and  if  D  is  the  midpoint  of  BG^  then  DE  is  equal  to 
DF. 

6.  If,  from  the  vertex  of  the  right  angle  of  a  triangle,  there  extend 
two  lines,  one  bisecting  the  hypotenuse,  and  the  other  perpendicular  to  it, 
the  angle  of  these  two  lines  is  equal  to  the  difference  between  the  two 
acute  angles  of  the  triangle. 

7.  In  the  base  of  a  triangle,  find  the  point  from  which  lines  extending 
to  the  sides,  and  parallel  to  them,  are  equal. 

8.  Can  two  unequal  triangles  have  a  side  and  two  angles  in  one  equal 
to  a  side  and  two  angles  in  the  other  ? 

9.  Of  all  triangles  on  the  same  base,  and  having  the  same  vertical 
angle,  the  isosceles  has  the  greatest  area. 

10.  The  lines  that  bisect  the  angles  formed  by  producing  the  sides 
of  a  quadrilateral  that  is  inscribed  in  a  circle  are  perpendicular  to  each 
other.     [IV,  26.] 

11.  Two  quadrilaterals  are  equivalent  when  their  diagonals  are  re- 
spectively equal  and  form  equal  angles.     [Compare  VI,  31,  Ex.  1.] 

12.  Lines  joining  the  midpoints  of  the  opposite  sides  of  any  quadri- 
lateral bisect  each  other. 

13.  In  the  triangle  ABC,  the  side  AB  =  13,  BG  =  15,  the  altitude  = 
12  ;  required  the  base  AG. 

14.  The  sides  of  a  triangle  have  the  ratio  of  65,  70,  and  75 ;  its  area 
is  21  square  inches ;  required  the  length  of  each  side. 

15.  The  area  of  a  triangle  that  has  one  angle  30°  is  one  fourth  the 
product  of  the  two  sides  containing  that  angle.  [Compare  V,  43,  Ex. 
11.] 

16.  An  inscribed  equilateral  triangle  has  one  fourth  the  area  of  a 
similar  circumscribed  triangle. 

17.  A  chord  is  8  inches,  and  the  altitude  of  its  segment  3  inches;  re- 
quired the  area  of  the  circle.     [V,  29,  o.] 

18.  What  is  the  area  of  the  sector  whose  arc  is  50°  and  whose  radius 
is  10  inches? 

19.  The  altitudes  of  a  triangle,  that  is,  the  perpendiculars  let  fall  from 
the  several  vertices  on  the  opposite  sides,  meet  in  one  point.  [Through 
each  vertex  of  the  triangle  make  a  line  parallel  to  the  opposite  side ;  and 
see  V,  2,  c] 


154 


PLANE  GEOMETRY. 


[Chap.  VIII. 


20.  If  the  medials  of  one  triangle  are  respectively  eqnal  to  the  sides  of 
a  second,  and  the  medials  of  the  second  are  respectively  equal  to  the 
sides  of  a  third,  the  third  is  similar  to  the  first.  What  is  the  linear 
ratio  ? 

21.  The  sum  of  twice  the  square  on  the  medial  of  a  triangle  and  twice 
the  square  on  half  the  base  is  equivalent  to  the  sum  of  the  squares  on  the 
other  two  sides.     [VI,  22  and  23.] 

22.  Three  times  the  sum  of  the  squares  on  the  sides  of  a  triangle  is 
equivalent  to  four  times  the  sum  of  the  squares  on  the  medials. 

23.  If  the  oblique  sides  of  a  trapezoid  are  produced  till  they  meet,  the 
point  of  meeting,  the  point  of  intersection  of  the  two  diagonals  of  the 
trapezoid,  and  the  midpoints  of  the  two  bases  are  all  in  one  straight  line. 
[Compare  V,  34  and  35.] 

24.  What  is  the  area  of  the  segment  whose  arc  is  36°  and  chord  6 
inches  ?     [This  chord  is  one  side  of  an  inscribed  decagon.] 

25.  The  sum  of  the  squares  on  the  sides  of  any  quadrilateral  is 
equivalent  to  the  sum  of  the  squares 

on  the  diagonals,  increased  by  four 
times  the  square  on  the  line  joining  the 
midpoints  of  the  diagonals.  [This  de- 
pends on  Ex.  22.] 

26.  If,  from  any  point  in  a  circum- 
ference, perpendiculars  fall  on  the 
sides  of  an  inscribed  triangle,  the 
three  points  of  intersection  are  in  the 
same  straight  line.  [From  /),  the 
perpendiculars  DE^  DF^  DG^  fall  on 
the  sides  of  the  triangle  ABG.^ 


15.  Exercises  in  Drawing. — 1.  Given  aline  divided  internally 
in  extreme  and  mean  ratio,  to  produce  it  to  the  point  where  it  is  divided 
externally  in  the  same  ratio,  without  using  any  other  lines. 

2.  From  two  points,  one  on  each  side  of  a  given  straight  Kne,  to  draw 
lines  making  an  angle  that  is  bisected  by  the  given  line. 

3.  To  describe  a  circumference  through  a  given  point,  and  touching 
two  given  straight  lines. 

4.  To  describe  four  equal  circumferences,  each  touching  two  of  the 
others  exteriorly,  and  all  touching  a  given  circumference  interiorly. 

5.  To  draw  lines  having  the  ratios  V  2  : 1,  -/  3  : 1,  V  5  : 1,  etc.  [V, 
38,  IX.] 

6.  To  draw  a  right  angle  by  means  of  VI,  23,  o. 


Art.  15.] 


CIRCLES, 


155 


7.  To  divide  a  circle  into  two  or  more  equivalent  parts  by  concentric 
circumferences. 

8.  To  divide  a  given  straight  line  in  the  ratio  of  the  areas  of  two 
given  squares.     [Compare  Euclidean  demonstration  of  VI,  28.] 

9.  To  construct  a  right-angled  triangle  when  the  area  and  hypote- 
nuse are  given. 

10.  Given  one  angle,  a  side  opposite  to  it,  and  the  difference  of  the 
other  two  sides,  to  construct  the  triangle.  [Let  CD  be  the  given  differ- 
ence, CB  the  side,  and  ^  the  angle.] 


c-^ 


11.  Given  one  angle,  a  side  opposite  to  it,  and  the  sum  of  the  other 
two  sides,  to  construct  the  triangle.  [Let  AB\)Q  the  side,  ^(7  the  sum, 
and  ADB  the  angle.] 

12.  To  construct  a  triangle  when  the  three  medials  are  given. 

13.  To  construct  a  triangle  when 
the  three  altitudes  are  given.  [Two 
sides  of  a  triangle  are  inversely  pro- 
portional to  the  corresponding  alti- 
tudes.] 

14.  To  construct  a  triangle,  when 
the  altitude,  the  medial,  and  the  line 
bisecting  the  vertical  angle  are  given. 
[Let  CD  be  the  altitude,  CE  the  bi- 
sector of  the  vertical  angle,  and  CF 
the  medial.] 


GEOMETRY   OF    SPACE. 


CHAPTER  IX. 
STRAIGHT  LINES  AND  PLANES. 

Article  1. — ^A  knowledge  of  the  properties  of  plane  figures 
is  the  basis  for  the  study  of  figures  that  do  not  lie  in  a  plane. 
The  student  who  understands  the  former  will  find  little  diffi- 
culty in  mastering  the  latter. 

Every  plane  figure  could  be  illustrated  by  the  diagram  on 
the  surface  of  the  paper.  That  is  not  always  possible  in  the 
sequel.  "Whenever  the  student  finds  it  difficult  to  see  the  figure, 
he  should  make  use  of  sticks,  wires,  or  threads,  to  represent 
lines,  and  of  sheets  of  paper  to  represent  surfaces.  These,  how- 
ever, are  only  helps  to  realize  the  perfect  figure  which  exists  in 
the  imagination. 

liines  in  Space. 

3.  The  definitions  of  angle  and  parallel  are  as  applicable  in 
space  as  to  plane  figures.  Also,  those  principles  concerning 
straight  lines  which  are  immediately  derived  from  the  Postulates 
and  Axioms  of  Geometry. 

Through  any  point  in  space  there  may  be  a  straight  line  in 
any  given  direction,  and  through  any  other  point  there  may  be 
one  and  only  one  straight  line  having  the  same  directions  as  the 
former.  Two  such  lines,  having  no  difference  in  direction,  are 
parallel  and  lie  in  one  plane. 

Any  other  line  that  is  parallel  to  one  of  these  two  is  also 


Art.  2,1  STRAIGHT  LINES  AND  PLANES,  157 

parallel  to  the  other.  The  three  may,  or  may  not,  lie  in  one 
plane.  There  may  be  any  number  of  parallel  lines  in  space,  no 
three  of  them  being  in  one  plane. 

Two  straight  lines  in  space  may  have  different  directions, 
and  two  such  lines  may  lie  in  one  plane,  or  they  may  not.  If 
they  lie  in  one  plane  they  meet  when  produced,  and  the  angle  is 
evident.  They  may  not  lie  in  one  plane,  as  for  example  the 
eastern  boundary  of  the  floor  and  the  northern  boundary  of  the 
ceiling.  Two  such  lines  can  never  be  produced  to  meet ;  yet 
they  are  not  parallel. 

The  amount  of  angle  of  two  straight  lines  that  do  not  lie  in 
one  plane  may  be  made  evident  by  a  third  line  parallel  to  one  of 
the  two  and  through  any  point  of  the  other.  There  is,  in  the 
third  line,  the  same  divergence  from  the  directions  of  the  par- 
allel not  met  as  from  those  of  the  parallel  that  is  met. 

CoroUary. — If  two  lines  are  respectively  parallel  to  two 
other  lines  in  space,  either  the  angle  of  the  first  two  is  equal  to 
the  angle  of  the  second  two,  or  they  are  supplementary. 

The  angles  are  equal  or  supplementary  as  in  the  case  of 
angles  in  one  plane  (III,  21  and  22). 

One  arm  of  an  angle  remaining  fixed  as  an  axis,  the  other 
may  rotate  about  it,  keeping  all  the  while  the  same  angle  with 
it.  Every  position  of  the  rotating  arm  is  in  fact  a  distinct  line. 
Thus  there  may  be  in  space  any  number  of  straight  lines  making 
the  same  angle  with  a  given  line,  and  meeting  it  at  the  same 
point,  but  the  axis  is  not  in  the  same  plane  as  any  two  of  the 
other  lines,  unless  the  two  are  on  directly  opposite  sides  of  the 
axis. 

Planes  and  Lines. 

3.  Theorem. — Three  points  that  are  not  in  one  straight  line 
fix  the  position  of  a  plane. 

That  is,  there  can  be  one  and  only  one  plane  through  three 
such  points.  It  has  been  explained  (II,  13)  how  there  can  be  a 
plane  through  any  three  points. 

If  there  can  be  two  planes  through  three  points,  as  A,  J?, 
and  C,  then  each  of  the  lines  ABy  BG,  and  AC,  lies  wholly  in 


158  GEOMETRY  OF  SPACE,  [Chap.  IX. 

both  of  the  supposed  planes  (II,  12) ;  that  is,  the  two  planes 
coincide  in  all  the  points  of  these  three  lines,  so  far  as  they  ex- 
tend. 

Then  they  must  coincide  in  the  surface  of  the  triangle  AB  (7, 
for  through  any  point  of  this  surface  a  straight  line  may  extend 
from  the  perimeter  on  one  side  and  be  produced  to  the  peri- 
meter beyond,  and  such  line  must  lie  in  both  planes.  But  two 
planes  that  coincide  to  any  extent  of  surface  must  coincide 
throughout,  and  are  in  fact  one  plane  (II,  11). 

Corollary. — The  position  of  a  plane  is  determined  by  any 
plane  figure,  except  a  straight  line. 

If  two  points  are  one  above  and  one  below  a  plane,  a  line 
joining  them  must  pass  through  the  plane  and  have  a  point  in  it, 
and  only  one  ;  for  a  straight  line  having  two  points  in  a  plane 
lies  wholly  in  it.  A  straight  line  and  a  plane  may,  therefore, 
have  a  single  point  in  common. 

When  a  line  and  a  plane  have  only  one  common  point,  the 
line  is  said  to  pierce  the  plane,  and  the  plane  to  cut  the 
line.  The  common  point  is  called  the  foot  of  the  line  in  the 
plane. 

When  a  line  lies  wholly  in  a  plane,  the  plane  is  said  to  pass 
through  the  line. 


4.  Theorem. — The  intersection  of  two  planes  is  a  straight 
line. 

One  plane  may  have  points  on  opposite  sides  of  another. 
For  there  can  be  a  plane  through  any  three  points. 

Two  such  planes  must  have  more  than  one  point  in  common. 
For,  in  the  first  plane,  there  may  be  a  circle  with  the  center  on 
one  side  and  part  of  the  circumference  on  the  other  side  of  the 
second  plane.  Several  radii  to  this  arc  must  pierce  the  second 
plane  at  several  points,  all  of  which  are  common  to  both  planes. 
Other  plane  figures,  such  as  a  triangle  or  other  polygon,  may  be 
used  to  demonstrate  that  the  two  planes  have  several  common 
points. 

All  these  points  lie  in  one  straight  line,  for  otherwise  the 
two  planes  would  be  one  (3). 


Art.  5.] 


STRAIGHT  LIFE8  AND  PLANES, 


159 


^h?-v 


Perpendicular  Lines. 

5o  Theorem. — A  straight  line  that  is  perpendicular  to  each 
of  two  straight  li?ies  at  their  poitit  of  intersection^  is  perpen- 
dicular to  every  straight  line  that  lies  in  the  plane  of  the  two, 
and  passes  through  their  point  of  intersection. 

In  the  diagram,  suppose  D,  B,  and  C  to  be  on  the  plane 
MN,   the  point  A  being  above, 
and  I  below  that  plane. 

If  the  line  AB  is  perpendicu- 
lar to  BG  and  to  BD,  it  is  to  be 
proved  that  it  is  also  perpen- 
dicular to  every  other  line  lying 
in  the  plane  MN",  and  passing 
through  the  point  B ;  as,  for 
example,  BE. 

Produce  AB,  making  BI 
equal  to  BA,  and  let  any  line,  as 

FH,  cut  the  lines  BC,  BE,  and  BD,  in  F,  G,  and  JI.     Then 
join  AF,  A  G,  AH,  and  IF,  IG,  and  IH. 

Now,  since  B  G  and  BB  are  perpendicular  to  AI  at  its  mid- 
point, the  triangles  AFH  and  IFH  have  ^i^  equal  to  IF  (III, 
10,  vii),  AH  equal  to  IH,  and  FH  common.  Therefore,  the 
angle  AHF  is  equal  to  IHF.  Then  the  triangles  AHG  and 
IHG  are  equal  (V,  11),  and  the  lines  AG  and  IG  are  equal. 
Therefore,  the  line  BE,  having  two  points  each  equally  distant 
from  A  and  I,  is  perpendicular  to  the  line  AI  (III,  11,  c). 

6.  Theorem. —  Conversely,  if  several  straight  lines  are  per- 
pendicular to  a  given  line  at  the  same  point,  these  several  lines 
lie  in  one  plane. 

If  BA  is  perpendicular  to 
BC,  to  BB,  and  to  BE,  it  is  to 
be  proved  that  these  three  all  lie 
in  one  plane. 

BD,  for  instance,  must  be  in 
the  plane  GBE.  For  the  inter- 
section of  the  plane  of  ABB  with 
the  plane  of  GBE  is  a  straight 


160  GEOMETRY  OF  SPACE.  [Chap.  IX 

line,  which  is  perpendicular  to  AJB  at 
the  point  B  (5).  Therefore,  it  coincides 
with  BB  (III,  9).  Thus,  any  line,  per- 
pendicular to  AB  at  the  point  J5,  is  in  the 
plane  of  (7,  B,  D,  and  E. 

This  theorem  is  the  ground  of  the  fol- 
lowing definition.  A  straight  line  is  said 
to  be  perpendicular  to  a  plane  when 
it  is  perpendicular  to  every  straight  line 

that  passes  through  its  foot  in  that  plane,  and  the  plane  is  said 
to  be  perpendicular  to  the  line. 

Corollaries. — I.  If  one  of  two  perpendicular  lines  revolves 
about  the  other,  the  revolving  line  describes  a  plane,  which  is 
perpendicular  to  the  axis. 

II.  Through  one  point  of  a  straight  line  there  can  be  only 
one  plane  perpendicular  to  that  line. 

III.  If  a  plane  cuts  a  straight  line  perpendicularly  at  the  mid- 
point of  the  line,  then  every  point  of  the  plane  is  equally  dis- 
tant from  the  two  ends  of  the  line. 

IV.  If  a  line  extends  from  the  foot  of  the  perpendicular  above 
the  plane,  it  makes  an  acute  angle  with  the  perpendicular  ;  and 
conversely. 

V.  The  locus  of  those  points  in  space  that  are  equally  distant 
from  two  given  points  is  the  plane  perpendicular  to  and  bisect- 
ing the  straight  line  joining  those  points  (III,  12). 


7.  Theorem. — Through  one  point  there  can  he  only  one  line 
perpendicular  to  a  plane. 

If  there  could  be  two,  these  two  lines  through  one  point 
would  determine  the  position  of  another  plane,  which  would  in- 
tersect the  given  plane.  Then  the  line  of  intersection  would  lie 
in  the  plane  of  the  supposed  perpendiculars,^nd  be  perpendicu- 
lar to  both  of  them. 

Thus  there  would  be  in  one  plane  two  perpendiculars  to  the 
same  line  through  one  point,  which  is  absurd.  Therefore,  the 
two  supposed  perpendiculars  to  a  plane  through  one  point  can 
not  exist. 


Art.  Y.] 


STRAIGHT  LINES  AND  PLANES. 


161 


Corollaries. — I.  If  one  of  two  parallel  lines  is  perpendicular 
to  a  plane,  the  other  is.  For,  to  every  line  in  the  plane  through 
the  foot  of  the  first  given  parallel,  there  may  be  in  the  plane  a 
parallel  line  through  the  foot  of  the  second  given  parallel.  The 
second  given  parallel  is  perpendicular  to  these  lines  (2,  c). 

II.  Conversely,  lines  perpendicular  to  one  plane  are  par- 
allel. 

III.  A  line  perpendicular  to  a  plane  is  perpendicular  to  every 
straight  line  in  the  plane  (2). 


Oblique  Liiues  and  Planes. 


8.  Theorem. — If,  from  a  point  without  a  plane,  a  perpen- 
dicular line  and  oblique  lines  extend  to  the  plane,  two  oblique 
lines  that  meet  the  plane  at  equal  distances  from  the  foot  of  the 
perpendicular  are  equal. 

Let  AB  be  perpendicular, 
and  -4  C  and  AD  oblique  to 
the  plane  MN,  and  the  dis- 
tances i?(7  and  BD  equal. 

Then  the  triangles  AB  O 
and  ABB  are  equal  (V,  11), 
and  AG  i^  equal  to  AB. 

Corollaries. — I.  A  per- 
pendicular is  the  shortest  line 

from  a  point  to  a  plane.     Hence,  the  distance  from  a  point  to  a 
plane  is  measured  by  a  perpendicular  line. 

II.  If  the  distances  from  the  perpendicular  are  unequal  the 
oblique  lines  are  unequal ;  and  the  greater  the  distance  the 
greater  the  line. 

The  Axis  of  a  Circle  is  the  straight  line  perpendicular  to 
the  plane  of  the  circle  at  its  center. 

HI.  All  points  of  the  circumference  of  a  circle  are  equidis- 
tant from  any  given  point  of  its  axis. 

IV.  The  locus  of  those  points  that  are  equally  distant  from 
three  given  points  is  the  axis  of  the  circle  whose  circumference 
passes  through  those  three  points. 
8 


162 


GEOMETRY  OF  SPACE. 


[Chap.  IX. 


•-S 


A^^' 


9.  If,  from  all  points  of  a  line,  perpendiculars  fall  on  a 
plane,  the  line  thus  described  on  the  plane  is  the  projection  of 
the  given  line  on  the  given  plane. 

Theorem. — The  projection  of  a  straight  line  on  a  plane  is 
a  straight  line. 

Let  AB  be  the  given  line, 
and  M  the  given  plane.  From 
the  point  B  let  the  perpendicu- 
lar BI)  fall  upon  the  plane. 

Every  perpendicular  to  M 
let  fall  from  a  point  of  AB 
must  be  parallel  to  BI),  and 
must  therefore  lie  in  the  plane 

AI),  and  meet  the  plane  M  in  some  point  of  the  intersection  of 
the  two  planes.  Hence  the  intersection  CD  is  the  projection  of 
the  straight  line  AB  on  the  plane  M. 

There  is  one  exception.  When  the  line  is  perpendicular  to 
the  plane,  its  projection  is  a  point. 

Corollary. — ^A  straight  line  and  its  'projection  on  a  plane 
both  lie  in  one  plane. 


M 


10.  Theorem. — The  angle  made  hy  a  straight  line  and  its 
projection  on  a  plane  is  smaller  than  the  angle  it  makes  with 
any  other  line  in  the  plane. 

Let  ^C  be  the  given  line,  and  BG  its  projection  on  the 
plane.      It    is    to    be    proved 
that  the   angle   ACB  is  less  ^ 

than  the  angle  made  hj  AG 
with  any  other  line  in  the  plane, 
as  GB. 

Make  AB  perpendicular  to 
the  plane.  Then  the  point  B 
is  in  the  projection  of  AG. 
Take  GB  equal  to  GB,  and 
join  AB. 

The  triangles  A  GB  and 
AGB  have  two  sides  of  one 
respectively  equal  to  two  sides  of  the  other,  and  the  third  side 


K 

• 

li"v\ 

H^nX 

|\    X 

\ 

I?'  ^ 

^^  f* 

\ . 

d'^ 

Art.  10.]  STRAIGHT  LINES  AND  PLANES.  163 

AI)  longer  than  the  third  side  AB.     Therefore,  the  angle  A  CD 
is  greater  than  the  angle  A  CB. 

The  angle  a  line  makes  with  its  projection  on  a  plane  is 
called  the  angle  of  the  line  and  plane. 


11.  Applications. — Three  points,  however  placed,  must  be  in  the 
same  plane.  It  is  on  tliis  principle  that  stability  is  more  readily  obtained 
by  three  supports  than  by  a  greater  number.  A  three-legged  stool  must 
be  steady. 

It  is  frequently  important  in  machinery  that  a  body  shall  have  what 
is  called  a  parallel  motion ;  that  is,  that  all  its  parts  shall  move  in  parallel 
lines.  ,  The  piston  of  a  steam-engine  and  the  rod  which  it  drives  receive 
such  a  motion. 

A  ready  way  of  constructing  a  line  perpendicular  to  a  plane  is  by  the 
use  of  two  squares  (III,  15).  Place  the  corner  of  each  at  the  foot  of  the 
desired  perpendicular,  one  edge  of  each  square  resting  on  the  plane  sur- 
face. Bring  their  perpendicular  edges  together.  The  position  of  these 
coincident  edges  must  be  that  of  a  perpendicular  to  the  plane,  for  it  is 
perpendicular  to  two  lines  in  the  plane. 

When  a  circle  rotates  on  its  axis,  the  figure  undergoes  no  change  of 
position,  each  point  of  the  circumference  taking  successively  the  position 
deserted  by  another  point. 

In  the  turning  lathe,  the  axis  on  which  the  material  is  made  to  rotate 
is  the  axis  of  the  circles  formed  by  the  cutting  tool,  as  it  removes  the 
matter  projecting  beyond  a  proper  distance  from  the  axis.  The  cross- 
section  of  every  part  of  the  thing  turned  is  a  circle,  all  the  circles  having 
the  same  axis. 


13,  Exercises. — 1.  Designate  two  lines  that  are  everywhere 
equally  distant,  but  which  are  not  parallel. 

2.  Designate  four  points  that  do  not  all  lie  in  one  plane. 

3.  Demonstrate  the  proposition  of  Article  5  without  using  lines  below 
the  plane  MN.  [Let  the  given  lines  in  the  plane  extend  beyond  B^  and 
construct  triangles  there.] 

4.  Two  equal  oblique  lines  from  the  same  point  make  equal  angles 
with  the  plane ;  also,  the  angles  they  make  with  the  perpendicular  are 
equal. 

5.  The  angle  that  a  line  makes  with  its  projection  produced  is 
larger  than  the  angle  made  with  any  other  line  in  the  plane. 


164 


GEOMETRY  OF  SPACE. 


[Chap.  IX. 


Diedrals. 

13,  A  Diedral  is  the  figure  formed  by  two  planes  which 
meet.  It  is  also  called  a  diedral  angle.  The  planes  are  its 
faces  and  the  intersection  is  its  edge. 

In  naming  a  diedral,  four  points  are  designated,  one  in  each 
face,  and  two  on  the  edge  ; 
the  letters  designating  the 
points  on  the  edge  are  placed 
between  the  other  two,  as 
the  diedral  BAEC  or  the 
diedral  (7^^2>.  When  there 
would  be  no  ambiguity,  a 
diedral  may  be  designated  by 
two  letters,  or  by  one. 

The  quantity  of  a  diedral  is  the  difference  of  direction  of  the 
two  faces  from  the  edge,  as  indicated  by  a  line  in  each  face  per- 
pendicular to  the  edge. 

Problem. — A  diedral  may  have  any  angular  quantity. 

The  plane  BE  may  be 
fixed  in  position,  also  the 
line  AE  and  the  line  FO 
perpendicular  to  AE.  A 
plane  may  rotate  from  the 
position  BEj  around  AE  as 
an  axis.  The  perpendicular 
describes  the  plane  EGO 
(6,  i).  The  difference  of 
direction  of  the  planes  BE 
and   CE  from  the  common 

line  AE  is  indicated  by  the  angle  EO  G.     Thus  the  diedral 
angle  BAEC  may  increase  from  zero  without  limit. 

Corollaries. — I.  A  diedral  may  be  equal  to  any  sum  or  to 
any  difference  of  angles,  or  it  may  have  any  ratio  to  another 
angle. 

II.  Diedral  angles  are  acute,  obtuse,  supplementary,  etc.,  as 
linear  angles  are. 

III.  All  right  diedral  angles  are  equal. 


Art.  13.] 


STRAIGHT  LINES  AND  PLANES. 


165 


IV.  When  two  planes  cut  each  other,  the  opposite  or  vertical 
diedral  angles  are  equal. 


Bight  Diedrals. 

14.  Theorem. — If  a  line  is  perpendicular  to  a  plane^  then 
any  plane  in  which  this  line  lies  is  perpendicular  to  the  other 
plane. 

If  AB  in  the  plane  P§  is  perpendicular  to  the  plane  M,  then 
AB  is  perpendicular  to  every  line  in 
M  which  passes  through  the  point 
B  ;  thus,  to  B  Q,  the  intersection  of 
the  two  planes,  and  to  B  O,  which  is 
made  perpendicular  to  the  intersec- 
tion BQ.  Then,  ABC  is  a  right 
angle.  But  the  diedral  angle  AB  Q  C 
is  the  same  as  the  linear  angle 
ABC.  ' '^ 

Corollaries. — I.  If  one  plane  is 
perpendicular  to  another,  a  straight  line  perpendicular  to  one  of 
them  at  some  point  of  their  intersection  must  lie  wholly  in  the 
other  plane  ;  for  there  may  be  in  the  second  plane  such  a  per- 
pendicular to  the  first  at  any  point  of  their  intersection,  and 
there  can  not  be  two  perpendiculars  to  the  same  plane  at  one 
point. 

II.  If  a  plane  is  perpendicular  to  another  plane,  a  line  in  one 
of  them  perpendicular  to  their  intersection  must  be  perpendicu- 
lar to  the  other  plane ;  for  it  is  perpendicular  to  two  lines  in 
that  other  plane. 

III.  If  two  planes  are  perpendicular  to  a  third,  the  intersec- 
tion of  the  first  two  is  a  line  perpendicular  to  the  third  plane. 


Oblique  Diedrals. 

15.  Theorem. — If  the  arms  of  an  angle  whose  vertex  is  on 
the  edge  of  a  diedral  are  respectively  perpendicular  to  the  faces 
of  the  diedral,  and  if  each  arm  is  on  the  same  side  of  the  plane 
to  which  it  is  perpendicular  as  the  other  face  is,  then  the  angle 
and  diedral  are  supplementary. 


166 


GEOMETRY  OF  SPACE. 


[Chap.  IX. 


The  plane  of  the  angle  GDH  is  perpendicular  to  both  faces 
of  the  diedral  (14),  and  to  the 
edge  AB  (14,  iii).  Therefore 
the  intersections  FD  and  DE 
of  this  plane  with  the  faces  of 
the  diedral  are  perpendicular 
to  AB,  and  the  angle  FDE  is 
the  same  as  the  diedral. 

Now,  as  the  angles  FDE 
and    GHD  lie  in   one   plane, 

their  sum  is  equal  to  the  sum  of  the  right  angles  FDH  and 
GDE. 

Corollary. — If  the  arms  of  an  angle  are  respectively  perpen- 
dicular to  the  faces  of  a  diedral,  the  angle  and  diedral  are  either 
equal  or  supplementary  ;  for  the  arms  are  respectively  parallel 
to  those  mentioned  in  the  theorem  (2,  c).     Compare  III,  25. 


16.  Theorem. — Every  point  of  a  plane  that  bisects  a  die- 
dral is  equally  distant  from  its  two  faces. 

Let  the  plane  EC  bisect  the  diedral  BB  CE.  It  is  to  be 
proved  that  every  point  of  this  plane,  as  A,  for  example,  is 
equally  distant  from  the  planes  B  C  and  EC. 

Pass  a  plane  through  A  perpendicular  to  B  C.  The  intersec- 
tions with  the  given  planes,  OH,  OA,  and  01,  are  all  perpen- 
dicular to  B  C.  Then  the  line 
AO  bisects  the  angle  HOI, 
and  the  point  A  is  equally  dis- 
tant from  the  lines  OH  and 
01  (III,  13).  But  the  distance 
of  A  from  these  lines  is  meas- 
ured by  the  perpendiculars  AH 
and  AI,  which  measure  its  dis- 
tances from  the  two  faces  B  C 
and  EC.     Therefore,  any  point 

of  the  bisecting  plane  is  equally  distant  from  the  two  faces  of 
the  given  diedral. 

Corollary. — The  locus  of  the  points  equally  distant  from  two 


Art.  16.]  STRAIGHT  LINES  AND  PLANES.  167 

given  planes  consists  of  two  planes  bisecting  the  adjacent  die- 
drals  formed  by  the  given  planes. 

17.  Applications. — The  theory  of  diedrals  is  as  important  in  the 
study  of  magnitudes  bounded  by  planes  as  the  theory  of  angles  in  the 
study  of  polygons. 

In  the  science  of  crystallography,  crystals  of  many  species  may  be 
classified  by  measuring  their  diedrals. 

The  plane  of  the  surface  of  a  liquid  at  rest  is  called  horizontal^  or  the 
plane  of  the  horizon.  The  direction  of  a  plumb-line  is  qsX\qA  vertical.  A 
vertical  plane  or  line  is  perpendicular  to  the  horizon,  the  positions  of  both 
being  governed  by  the  same  causes. 

Horizontal  and  vertical  planes  are  in  frequent  use.  Floors,  walls, 
ceilings,  etc.,  are  examples.  The  methods  of  using  the  builder's  level  and 
plummet  are  among  the  simplest  applications  of  geometrical  principles. 
Civil  engineers,  astronomers,  and  navigators  refer  to  the  horizon,  or  to  a 
vertical  plane,  in  their  observations. 

18.  Exercises. — 1.  If  a  line  is  perpendicular  to  a  plane,  and  if 
from  its  foot  a  perpendicular  falls  on  some  other  line  which  lies  in  the 
plane,  then  this  last  line  is  perpendicular  to  the  plane  of  the  other  two. 

2.  What  is  the  locus  of  those  points  in  space,  each  of  which  is 
equally  distant  from  two  given  straight  lines  that  lie  in  the  same 
plane  ? 

3.  What  is  the  locus  of  those  points  in  space,  each  of  which  is 
equally  distant  from  three  given  straight  lines  that  lie  in  the  same 
plane  ? 

4.  "What  is  the  locus  of  those  points  in  space,  such  that  the  sum  of 
the  distances  of  each  from  two  given  planes  is  equal  to  a  given  straight 
line? 

Parallel  Lines  and  Planes. 

19.  Parallelism  consists  in  the  identity  of  the  directions 
of  lines,  or  of  a  line  and  a  plane,  or  of  planes. 

A  Parallel  Line  and  Plane  are  such  that  the  line  is  par- 
allel to  a  line  in  the  plane  ;  that  is,  the  plane  has  directions 
which  are  the  same  that  the  line  has. 

Corollaries. — I.  When  a  line  and  a  plane  are  parallel,  there 
may  be  through  any  point  of  the  plane  a  line  parallel  to  the 
given  line. 


168  GEOMETRY  OF  SPACE.  [Chap.  IX. 

II.  A  line  that  is  parallel  to  a  plane  is  parallel  to  its  projec- 
tion on  that  plane. 

III.  A  line  parallel  to  a  plane  is  everywhere  equally  distant 
from  it. 

Parallel  Planes. 

30.  Parallel  Planes  are  such  that  every  straight  line  in 
one  has  a  parallel  line  in  the  other  ;  that 
is,  the  planes  have  the  same  directions. 
If  the  parallel  lines  AB  and  CD  re- 
volve about  the  line  EF^  to  which 
they  are  both  perpendicular,  then  each 
of  the  revolving  lines  describes  a  plane. 
Every  direction  assumed  by  one  line 
is  the  same  as  that  of  the  other,  and 
in  the  course  of  a  revolution  they  take 
all  the  directions  of  the  two  planes. 

Corollaries. — I.  Two  planes  parallel  to  a  third  are  parallel 
to  each  other. 

II.  Two  planes  perpendicular  to  the  same  straight  line  are 
parallel  to  each  other. 

III.  A  straight  line  perpendicular  to  one  of  two  parallel 
planes  is  perpendicular  to  the  other. 

ly.  In  one  of  two  parallel  planes,  and  through  any  point  of 
it,  there  may  be  a  straight  line  parallel  to  any  straight  line  in  the 
other  plane. 

V.  Parallel  planes  can  not  meet ;  for,  if  they  had  a  common 
point,  there  could  be  two  straight  lines  in  one  direction  from 
that  point. 

21.  Theorem. — The  intersections 
of  two  parallel  planes  hy  a  third  plane 
are  parallel  lines. 

Since  the  planes  3f  and  N"  are  par- 
allel, the  intersections  AB  and  CD 
can  never  meet.  Since  AB  and  CD 
lie  in  one  plane,  P,  and  can  never  meet, 
they  are  parallel  (III,  29,  c). 


.f] 

V 

__.  i\    / 

~UI 

y 

N    / 

Art.  22.] 


STRAIGHT  LINES  AND  PLANES. 


169 


33.  Theorem, — The  parts  of  parallel  lines  intercepted  be- 
tween parallel  planes  are  equal. 

The  lines  AB  and  CD,  being 
parallel,  lie  in  one  plane.  Then 
AG  and  J3D,  the  intersections 
of  this  plane  with  the  parallel 
planes  M  and  P,  are  parallel 
lines.  Therefore,  AB  is  equal 
to  CD  (VI,  3). 


M  Ar C 

^         B n 


23.  Theorem. — Two  parallel  planes  are  everywhere  equally 
distant. 

For  the  distance  is  measured  by  perpendiculars  (8,  i). 

Such  perpendicular  lines  are  parallel  (7,  ii),  and  therefore 
equal. 


^B 


24.  Theorem. — If  the  arms  of  an  angle  are  both  parallel 
to  a  given  plane,  the  plane  of  that  angle  is  parallel  to  the  given 
plane. 

Let  BA  C  be  the  given  angle 
and  31  the  given  plane. 

Make  AB  perpendicular  to 
iHf,  and  make  BE  and  BF  in 
that  plane  respectively  parallel 
to  AB  and  A  C  (19,  i).  Then 
BA,  being  perpendicular  to  BE 
and  to  BF,  is  perpendicular  to 
AB  and  io  AG  (III,  22,  v),  and 
to  the  plane  of  BA  G ;  and  the 
two  planes  are  parallel  (20,  ii). 


25.  Theorem.. — If  two  straight  lines  that  cut  each  other 
are  respectively  parallel  to  two  other  straight  lines  that  cut  each 
other,  then  the  plane  of  the  first  two  is  parallel  to  the  plane  of  the 
second  two. 


170 


GEOMETRY  OF  SPACE. 


[Chap.  IX. 


Let  AB  be  parallel  to  EF, 
and  CD  parallel  to  GIT, 

Both  AB  and  CB  must 
be  parallel  to  the  plane  N', 
since  they  are  parallel  to  lines 
in  that  plane.  Therefore  the 
plane  31  is  parallel  to  iV^ 


26.  Theorem, — Straight  lines  cut  hy  three  parallel  planes 
are  divided  proportionally. 

The  line  AB  is  cut  at  ^, 
E,  and  B,  and  the  line  GB  is 
cut  at  (7,  F,  and  B,  by  the 
parallel  planes  Jf,  iV,  and  P. 
Joining  AB,  it  pierces  the 
plane  iV  at  (r.  Then  the 
plane  ABB  makes  the  inter- 
sections FG  and  BB  with  the 
planes  iVand  P,  and  the  plane 
ABC  makes  the  intersections 
A  C  and  GF  with  the  planes 
JIf  and  J^. 

Since  FG  is  parallel  to  BB  (21), 


AF'.FB=^AG\GB. 


For  a  like  reason, 

^6?:  GB=  CF'.FB. 
Therefore,  AF  xFB=CF'.  FB. 


Corollary. — The  segments  are  proportional  to  the  distances 
between  the  planes,  that  is,  AF  is  to  FB  as  the  distance  M — 
2^  is  to  the  distance  K — P. 


27.  Applications. — The  general  problem  of  perspective  in  draw- 
lug  consists  in  representing  upon  a  plane  surface  the  apparent  form  of 
objects  in  sight.     This  plane,  the  plane  of  the  picture,  is  supposed  to  be 


Art.  27.]  STRAIGHT  LINES  AND  PLANES.  171 

between  the  eye  and  the  objects  to  be  drawn.  Then  each  object  is  to  bo 
represented  upon  the  plane,  at  the  point  where  it  is  pierced  by  the  visual 
ray  from  the  object  to  the  eye. 

All  the  visual  rays  from  one  straight  object,  such  as  the  top  of  a  wall, 
or  one  corner  of  a  house,  lie  in  one  plane.  Their  intersection  with  the 
plane  of  the  picture  must  be  a  straight  line.  Therefore,  all  straight  ob- 
jects, whatever  their  position,  are  drawn  as  straight  lines. 

If  parallel  straight  objects  are  also  parallel  to  the  plane  of  the  picture, 
they  are  drawn  parallel,  for  the  lines  drawn  must  be  parallel  to  the  ob- 
jects, and  therefore  to  each  other. 

Two  parallel  objects,  which  are  not  parallel  to  the  plane  of  the  pict- 
ure, are  not  parallel  in  the  perspective.  The  lines  meet,  if  produced,  at 
that  point  where  the  plane  of  the  picture  is  pierced  by  a  line  from  the 
eye  parallel  to  the  objects. 


38.  Exercises. — 1.  A  straight  line  makes  equal  angles  with  two 
parallel  planes. 

2.  The  projections  of  two  parallel  lines  on  a  plane  are  parallel. 

3.  Two  parallel  lines  make  equal  angles  with  a  given  plane. 

4.  When  two  planes  are  both  perpendicular  to  a  third,  and  their  in- 
tersections with  the  third  plane  are  parallel  lines,  the  two  planes  are  par- 
allel to  each  other.  [Make  a  line  in  the  third  plane  perpendicular  to  the 
intersections.] 

5.  Given  any  two  straight  lines  in  space;  either  one  plane  may  pass 
through  both,  or  two  parallel  planes  may  pass  through  them,  that  is, 
one  through  each  point. 

6.  Demonstrate  the  last  sentence  of  Article  27.  [If  two  faces  of  a 
diedral  have  a  Kne  in  one  parallel  to  a  line  in  the  other,  both  these  lines 
are  parallel  to  the  edge  of  the  diedral.] 


172 


GEOMETRY  OF  SPACE. 


[Chap.  IX. 


Triedrals. 


29.  When  three  planes 
cut  each  other,  three  cases 
are  possible. 

1st.  The  intersections 
may  coincide.  Then  six  die- 
drals  are  formed,  having  for 
their  common  edge  the  in- 
tersection of  the  planes. 

2d.  The    three    intersec- 
tions may  be  parallel  lines, 
intersection  of  the  other  two. 


Then  each  plane  is  parallel  to  the 


3d.  The  three  intersections  may  meet  at  one  point.  Then 
the  space  about  the  point  is  divided  by  the  three  planes  into 
eight  parts. 

Notice  that  two  intersecting  planes  make  four  diedrals,  and 
a  third  plane  divides  each  diedral  into  two  parts.  Each  of  these 
parts  is  called  a  triedral. 

A  fourth  case  is  impossible.  For,  since  any  two  of  the  in- 
tersections lie  in  one  plane,  either  they  are  parallel,  or  they  meet. 
If  two  intersections  meet,  the  point  of  meeting  is  common  to 
the  three  planes,  and  therefore  common  to  all  the  intersections. 


Art.  29.]  STRAIGHT  LINES  AND  PLANES.  173 

Hence,  either  the  three  intersections  coincide,  or  they  are  par- 
allel, or  they  have  only  one  common  point.  But  these  are  the 
three  cases  just  considered. 

A  Triedral  is  the  figure  formed  by  three  planes  meeting  at 
one  point.  The  point  where  the  planes  and  intersections  all 
meet  is  called  the  vertex  of  the  triedral.  The  intersections 
are  its  edges,  and  the  angles  formed  by  the  edges  are  its 
faces. 

The  planes  of  the  angles  are  sometimes  called  faces.  Al- 
ways when  the  term  face  designates  a  quantity,  it  means  the 
angles. 

The  corners  of  a  room,  or  of  a  chest,  are  illustrations  of  trie- 
drals  with  rectangular  faces.  The  point  of  a  triangular  file, 
or  of  a  small-sword,  has  the  form  of  a  triedral  with  acute 
faces. 

The  triedral  has  many  things  analogous  to  the  plane  trian- 
gle. It  has  been  called  a  solid  triangle  ;  and  more  frequently, 
but  with  less  propriety,  a  solid  angle.  The  three  planes  make 
three  diedrals.  The  diedrals  and  the  faces  are  the  six  elements 
of  a  triedral.     They  are  all  angular  quantities. 

A  triedral  that  has  one  rectangular  diedral  is  called  a  rect- 
angular triedral.  If  it  has  two,  it  is  birectangular ;  if  it 
has  three,  it  is  trirectangular. 

A  triedral  that  has  two  of  its  faces  equal  is  called  isos- 
celes ;  if  all  three  are  equal,  it  is  equilateral. 

A  triedral  may  be  named  by  a  single  letter  at  the  vertex,  but, 
when  two  triedrals  have  the  same  vertex,  add  three  letters,  one 
on  each  edge,  thus  :  S — ABC, 

Supplementary  Triedrals. 

30.  Theorem. — If,  at  the  vertex  of  a  triedral,  there  is  a 
perpendicular  to  the  plane  of  each  face,  and  on  the  same  side  of 
the  plane  as  the  other  parts  of  the  triedral,  the  three  perpendicu- 
lars are  the  edges  of  a  second  triedral ; 

Each  edge  of  the  first  triedral  is  perpendicular  to  the  plane 
of  (^  fctce  of  the  second,  and  on  the  same  side  of  it  as  the  other 
parts  of  the  figure  ; 


174  GEOMETRY  OF  SPACE.  [Chap.  IX. 

^Each  diedral  of  the  first  is  a  supplement  of  the  opposite  face 
of  the  second  triedral ;  and 

Each  face  of  the  first  is  a  supplement  of  the  opposite  diedral 
of  the  second. 

Let  S—ABC  be  the  first  trie- 
dral,  SF  perpendicular  to  the  plane  C  \ 

A  SB  and  on  the  same  side  of  it  as  > 

SO  J  jSI)  perpendicular  to  the  plane 
CjSB  and  on  the  same  side  of  it  as  ^^^:^^rr~-— _^ 

/SAy  and  jSF  perpendicular  to  the  ^^^ /         ^-^.^ 

plane  ASC  and  on  the  same  side  of  -^       /  ^ 

it  as  SB.     Then  S—BEF  is  the  '"^ 

second  triedral. 

Since  SF  is  perpendicular  to  the  plane  ASB^  it  is  perpendic- 
ular to  the  line  8A.  Since  SE  is  perpendicular  to  the  plane 
CSA,  it  is  perpendicular  to  SA.  Then  SA,  being  perpendic- 
ular to  the  two  lines  SF  and  SE,  is  perpendicular  to  the  plane 
FSE.  Since  SB  and  SA  are  on  the  same  side  of  the  plane 
CSB  the  angle  A  SB  is  acute  (6,  iv)  ;  and  SB  and  SA  are  on 
the  same  side  of  the  plane  FSE. 

In  the  same  manner,  it  is  proved  that  SB  is  perpen- 
dicular to  the  plane  FSB  and  on  the  same  side  of  it  as 
SE'y  also  that  SG  i^  perpendicular  to  BSE  and  on  the  same 
side  of  it  as  SF. 

Since  SF  and  SE  are  perpendicular  to  SAy  the  diedral 
whose  edge  is  SA  is  the  supplement  of  the  face  ESF  (15). 
Similarly,  the  diedral  SB  is  the  supplement  of  the  face  BSF 
and  the  diedral  SC  \^  the  supplement  of  the  face  BSE.  Also, 
the  faces  A  SB,  ASC,  and  BSC,  are  respectively  supplements 
of  the  diedrals  SF,  SE,  and  SB. 

Supplementary  Triedrals  are  two  triedrals,  in  which  the 
faces  and  diedrals  of  the  one  are  respectively  the  supplements  of 
the  diedrals  and  faces  of  the  other.  Two  triedrals  can  have  this 
reciprocal  relation  without  having  a  common  vertex.  This  posi- 
tion is  used  here  for  purposes  of  demonstration  only. 


Art.  31.] 


STRAIGHT  LINES  AND  PLANES. 


175 


Faces  of  a  Triedral. 


31.  Theorem. — Each  face  of  a  triedral  is  less  than  the  sum 
of  the  other  two. 

The  theorem  is  demonstrated,  when  it  is  shown  that  the 
greatest  face  is  less  than  the  sum  of  the  other  two. 

Let  ASB  be  the  greatest  of  the  three  faces  of  the  triedral 
S.  From  the  face  ASB  take  the  part  ASD,  equal  to  the  face 
ASC.  Join  the  edges  SA  and  SB  by  any  straight  line  AB, 
Take  SC  equal  to  SB,  and  join  ^  (7  and  ^  C. 

Since  the  triangles  ASB  and  ASC  are  equal  (V,  11),  AB 
is  equal  to  A  C.  But  AB  \i  less 
than  the  sum  oi  AG  and  BG, 
and  from  these,  subtracting  the 
equals  AB  and  AG,  we  have 
BB  less  than  BG.  Hence,  the 
triangles  BSB  and  BSG  have 
two  sides  of  the  one  equal  to  two 
sides  of  the  other,  and  the  third 
side  BB  less  than  the  third  side 

B  G.  Therefore,  the  included  angle  BSB  is  less  than  the  angle 
BSG.  Adding  to  these  the  equal  angles  ASB  and  ASG,wq 
have  the  face  ASB  less  than  the  sum  of  the  faces  ASG  and 
BSG. 


33.  Theorem.— The  sum  of  the  faces  of  a  triedral  is  less 
than  four  right  angles. 

Through  any  three  points,  one  in  each  edge  of  the  triedral, 
let  the  plane  AB  G  pass,  making  the  intersections  ABy  B  G,  and 
A  G,  with  the  faces. 

There  is  formed  a  triedral 
at  each  of  the  points  A,  B,  and 
G.  Then  the  face  BA  G  is  less 
than  the  sum  of  BAS  and  GAS, 
the  face  ABG  is  less  than  the 
sum  of  ABS  and  GBS,  and  the 
face  B  GA  is  less  than  the  sum  of 


176  GEOMETRY  OF  SPACE.  [Chap.  IX. 

A  CS  and  B  CS.     Adding,  we  find  that  the  sum  of  the  angles 

of  the  triangle  ABC,  which  is 

two  right  angles,  is  less  than  the  ^ 

sum    of    the   six   angles   at   the  A\ 

bases  of  the  triangles  whose  ver-  /    \  \ 

tices  are  at  /SL  /  \    \ 

The  sum  of  all  the  angles  of  ^  /  \      \^ 

these  three  triangles  is  six  right  /       W'     \ 

angles.    Therefore,  since  the  sum 

of  those  at  the  bases  is  more  than 

two  right  angles,  the  sum  of  those  at  the  vertex  S  is  less  than 

four  right  angles. 

Take  any  three  points  on  the  paper  or  blackboard  for  A,  B, 
and  C.  Take  S  at  some  distance  from  the  surface,  so  that  the 
angles  formed  at  S>  are  quite  acute.  Then  let  &  approach  the 
surface  of  the  triangle  AB  C.  The  angles  at  S  increase  until 
the  point  S  reaches  the  surface  of  the  triangle,  when  the  sum 
becomes  four  right  angles,  and  the  triedral  becomes  a  plane. 

Sum  of  tlie  Diedrals. 

33.  Theorem. — In  every  triedral  the  sum  of  the  diedrals  is 
greater  than  two  right  angles,  and  less  than  six. 

Consider  the  supplementary  triedral,  with  the  given  one. 
Now,  the  sum  of  the  diedrals  of  the  given  triedral,  and  of  the 
faces  of  its  supplementary  triedral,  is  six  right  angles  ;  for  the 
sum  of  each  pair  is  two  right  angles.  But  the  sum  of  the  faces 
of  the  supplementary  triedral  is  less  than  four  right  angles,  and 
is  greater  than  zero.  Subtracting  this  sum  from  the  former,  the 
remainder,  being  the  sum  of  the  diedrals  of  the  given  triedral,  is 
greater  than  two  and  less  than  six  right  angles. 

Symmetrical  Triedrals. 

34,  Symmetrical  Triedrals  are  two  triedrals  whose  ele- 
ments are  respectively  equal,  but  arranged  in  reverse  order. 

If  the  edges  of  a  triedral  are  produced  beyond  the  vertex, 
they  form  the  edges  of  a  new  triedral.  The  faces  of  these  two 
triedrals  are  respectively  equal,  for  the  angles  are  vertical. 


Art.  34.]  STRAIGHT  LINES  AND  PLANES,  177 

Thus,  the  angles  ASO  and  ESD  are  equal ;  also,  the  angles 
JBSO  and  FSE  are  equal,   and 
the  angles  A8B  and  BSF. 

The  diedrals  whose  edges  are 
FS  and  BS  are  also  equal,  since, 
being  formed  by  the  same  planes, 
FFSBG  and  DFSBA,  they  are 
vertically  opposite  diedrals.  The 
same  is  true  of  the  diedrals  whose 
edges  are  BS  and  SA,  and  of  the 
diedrals  whose  edges  are  FS  and 
SO. 

The  point  ^iS'  is  a  center  of  symmetry,  and  the  triedrals  are 
symmetrical  with  reference  to  this  center  (IV,  5).  Symmetry 
with  reference  to  a  point,  line  or  plane  is  symmetry  of  posi- 
tion, but  triedrals  such  as  defined  in  the  first  sentence  of  this 
article  need  not  have  a  common  vertex.  Their  symmetry  is 
called  the  symmetry  of  form.  This  distinction  was  not 
needed  in  plane  geometry,  for  symmetrical  plane  figures  are 
equal. 

Two  points  are  symmetrical  with  reference  to  a  plane  Ol 
symmetry  when  the  line  that  joins  them  is  bisected  perpen- 
dicularly by  the  plane.  Two  figures,  or  two  parts  of  one  figure, 
are  symmetrical  with  reference  to  a  plane  of  symmetry,  when 
every  point  in  one  has  its  symmetrical  point  in  the  other. 

The  diagram  in  Article  5  of  this  chapter  illustrates  a  figure 
whose  two  parts  are  symmetrical  with  reference  to  the  plane  of 
symmetry  MIST,  Every  point  in  the  lines  AB,  AH,  AF,  has 
its  symmetrical  point  in  the  lines  IB,  IH,  IF. 

Models  of  symmetrical  triedrals  may  be  made  as  follows. 
Draw  four  lines  from  a  common 
point  A,  making  three  angles, 
BAC,  GAB,  and  BAE,  which 
are  unequal,  neither  one  larger 
than  the  sum  of  the  other  two, 
and  the  sum  of  the  three  less  than 
four  right  angles.  Cut  out  the 
remaining  angle  BAE.     Fold  the 


178  GEOMETRY  OF  SPACE,  [Chap.  IX. 

paper  at  the  lines  AG  and  AD  toward  the  obverse  face,  and 
bring  the  edges  AE  and  AJ3  together. 

Make  a  second  model  with  angles  equal  to  these,  cut,  and 
fold  toward  the  reverse  face. 

These  two  models  may  be  placed  vertex  to  vertex,  with  the 
equal  angles  vertically  opposite,  as  in  the  diagram.  Also  they 
may  be  placed  with  a  face  of  one  against  the  equal  face  of  the 
other.  Then  these  faces  illustrate  a  plane  of  symmetry.  This, 
however,  depends  on  a  principle  to  be  demonstrated  (35). 

A  triedral  having  three  unequal  faces  is  not  equal  to  its  sym- 
metrical, for  they  can  not  be  made  to  coincide.  The  elements 
are  arranged  in  reverse  order.  This  is  evident,  if  an  attempt  is 
made  to  insert  one  of  the  models  in  the  other.  Symmetry  is 
also  illustrated  by  two  gloves,  which  are  composed  of  equal 
parts  arranged  in  reverse  order. 

Equality  of  Triedrals. 

35.  Equality  may  be  shown  by  the  possible  coincidence  of 
the  respective  faces,  for  then  the  edges  must  coincide  ;  or,  by 
the  possible  coincidence  of  the  edges,  for  then  the  faces  must 
coincide. 

Theorem, —  When  two  triedrals  have  th^ir  faces  respectively 
equaly  their  diedrals  are  respectively  equal. 


Suppose  the  faces  ASB  and  DTE  equal,  also  ASC  and 
DTF,  also  DSC  and  ETF. 

On  the  several  edges  take  equal  lengths,  SA,  SB,  SC,  TD, 
TE,  and  TF\  join  AB,  AC,  BC,  DE,  DF,  and  EF  The 
isosceles  triangles  ASB  and  DTE  are  equal  (V,  11)  ;  so  are 


Art.  35.]  STRAIGHT  LINES  AND  PLANES.  179 

ASG  and  DTF\  also  JBSG  and  JETF.  Hence  the  triangles 
AB  C  and  DFF  are  equal. 

From  any  point  on  AS  make  MN"  and  MP,  both  perpendic- 
ular to  AS,  and  in  the  planes  ASB  and  ^/S'(7  respectively. 
Since  SAB  is  an  angle  at  the  base  of  an  isosceles  triangle,  it  is 
acute,  and  MN"  must  meet  AB.  For  a  like  reason,  MP  must 
meet  A  C.  Join  iVP.  Take  7>  (x  equal  to  AM  and  repeat  in 
this  triedral  the  same  construction. 

The  triangles  AMN'  and  J)  GK  are  equal  (V,  12,  c),  and 
AN  is  equal  to  DK.  For  a  like  reason,  AP  is  equal  to  DH. 
The  triangles  ^iVP  and  DKH  are  equal  (V,  11),  and  NP  is 
equal  to  KH.  Then  the  triangles  MNP  and  GKH  are  equal 
(V,  10),  and  the  angles  NMP  and  KGH  are  equal,  that  is,  the 
diedrals  >S14.  and  TB  are  equal.  In  the  same  manner,  it  is  shown 
that  the  diedrals  SB  and  TF  are  equal ;  also  the  diedrals  SG 
and  TF 

36.  Theorem. —  Two  triedrals  that  have  their  diedrals  re- 
spectively equal  have  their  faces  respectively  equal. 

Let  G  and  g  represent  the  given  triedrals,  and  S  and  s  their 
respective  supplementary  triedrals.  The  faces  of  S  are  respec- 
tively equal  to  those  of  s,  for  they  are  the  supplements  of  die- 
drals that  are  equal  by  hypothesis.  Therefore  the  diedrals  of  S 
are  respectively  equal  to  that  of  s  (35).  But  the  faces  of  G  and 
of  g  are  respectively  the  supplements  of  the  diedrals  of  aS'  and  of 
s.     Therefore  these  faces  are  respectively  equal. 

37.  Theorem. — Two  triedrals  that  have  two  faces  and  the 
included  diedral  of  one  respectively  equal  to  the  corresponding 
elements  of  the  other  have  the  remaining  elements  respectively 
equal. 

There  are  two  cases  to  be  considered. 

1st.  Suppose  the  angles 
AEO  and  BG  G  equal,  and  the 
angles  AEI  and  B  GD  equal, 
also  the  included  diedrals 
whose  edges  are  AE  and  B  G. 
Let  the   arrangement  be  the 


180 


GEOMETRY  OF  SPACE. 


[Chap.  IX. 


same  in  both,  so  that,  if  we  go  around  one  triedral  in  the  order 
O,  Ay  I,   O,  and  around  the  other  in  the  order  G,  JB,  J9,  G, 
in  both  cases  the  triedral  is 
on  the  right. 

Place  the  face  BCD  di- 
rectly upon  its  equal,  AEL 
Since  the  diedrals  are  equal, 
and  are  on  the  same  side  of 
the  plane  AEI,  the  planes 
BOG  and  AEO  coincide. 

Since  the  faces  BCG  and  AEO  are  equal,  the  lines  CG  and 
£10  coincide.  Thus,  the  angles  DCG  and  lEO  coincide,  and 
the  two  triedrals  coincide  throughout. 

2d.  Let  the  angles  AEO  and  BOG  he  equal,  and  the  angles 
AEI  and  B  CB,  also  the  included  diedrals,  whose  edges  are  AE 
and  B  C.  But  let  the  arrangement  be  the  reverse  ;  that  is,  if  we 
go  around  one  triedral  in  the  order  0,  Ay  ly  Oy  and  around  the 
other  in  the  order  (r,  i>,  By  Gy  in  one  case  the  triedral  is  to  the 
right,  and  in  the  other  to  the  left.  Then  it  may  be  proved  that 
the  two  triedrals  are  symmetrical. 

One  of  the  triedrals  can  be  made  to  coincide  with  the  sym- 
metrical of  the  other  ;  for,  if  the  edges  BG,  GCy  and  B  G  are 
produced  beyond  (7,  the  triedral  G — EBEhas  two  faces  and 
the  included  diedral  respect- 
ively equal  to  those  parts  of 
the  triedral  E —  A  Oly  and 
arranged  in  the  same  order  ; 
that  is,  the  reverse  of  the 
triedral  G—BGB.  Hence, 
as  just  shown,  the  triedrals 
G  —  FHK  and  E^AOI 
are  equal.     Therefore,  E — 
A 01  and   G—BGB  are 
symmetrical  triedrals. 

In  both  cases,  all  the  ele- 
ments of  one  triedral  are  respectively  equal  to  those  of  the 
other. 


\Kr 


jt-^ 


/ 


Art.  38.]  STRAIGHT  LINES  AND  PLANES,  181 

38#  Theorem. — Two  triedrals  that  have  one  face  and  the 
two  adjacent  diedrals  of  one  respectively  equal  to  the  correspond- 
ing elements  of  the  other,  have  the  remaining  elements  respective- 
ly equal. 

If  the  arrangement  of  the  corresponding  elements  is  the  same 
in  both,  the  equality  of  the  triedrals  may  be  demonstrated  by 
superposition.  If  the  arrangement  in  one  is  the  reverse  of  the 
other,  the  equality  of  the  remaining,  elements  may  be  demon- 
strated by  the  symmetrical  triedral,  as  in  the  preceding  article. 

This  theorem  may  also  be  demonstrated  by  the  principle  of 
supplementary  triedrals  (30).  Let  G  and  g  represent  the  given 
triedrals,  and  S  and  s  their  respective  supplementary  triedrals. 
Since  G  and  g  have  one  face  and  two  adjacent  diedrals  of  one 
respectively  equal  to  the  corresponding  elements  of  the  other,  S 
and  s  must  have  two  faces  and  the  included  diedral  of  one  re- 
spectively equal  to  the  corresponding  elements  of  the  other. 
Therefore  (37),  S  and  s  have  their  remaining  elements,  one  face 
and  the  adjacent  diedrals,  respectively  equal.  Then  the  remain- 
ing elements  of  G  and  g,  being  the  supplements  of  these,  must 
be  respectively  equal. 

Corollaries. — I.  In  all  cases  where  two  triedrals  have  all 
their  elements  respectively  equal,  if  the  arrangement  is  the  same 
the  triedrals  are  equal,  and  if  the  arrangement  is  the  reverse  the 
triedrals  are  symmetrical. 

II.  An  isosceles  triedral  and  its  symmetrical  are  equal  (37). 

III.  In  an  isosceles  triedral,  the  diedjals  opposite  the  equal 
faces  are  equal. 

IV.  If  two  diedrals  of  a  triedral  are  equal,  the  faces  opposite 
these  diedrals  are  equal. 

39.  Exercises. — 1.  Of  two  unequal  diedrals  of  a  triedral  the  face 
opposite  the  greater  is  greater  than  the  face  opposite  the  less ;  and  con- 
versely. 

2.  What  is  the  locus  of  those  points  in  space,  each  of  which  is  equally 
distant  from  three  given  planes  ? 


182 


GEOMETRY  OF  SPACE. 


[Chap.  IX. 


Polyedrals. 

40.  A  Polyedral  is  a  figure  formed  by  several  planes  that 
meet  at  one  point.  The  vertex,  edges,  and  faces  are  defined,  as 
those  of  a  triedral.  A  triedral  is  a  polyedral  formed  by  three 
planes. 

If  the  edges  of  a  polyedral  are  produced  beyond  the  vertex, 
they  form  the  edges  of  a  second  polyedral,  symmetrical  to  the 
former. 

Several  properties  of  triedrals  are  common  to  other  polye- 
drals. 

Problem. — Any  polyedral  of  more  than  three  faces  may  he 
divided  into  triedrals. 

For  a  plane  may  pass  through  any  two  edges  that  are 
not  consecutive.  Thus,  a  polyedral  of  four  faces  may  be 
divided  into  two  triedrals  ;  one  of  five  faces,  into  three,  and 
so  on. 

This  is  like  the  division  of  a  polygon  into  triangles.  The 
plane  passing  through  two  edges  not  consecutive  is  called  a 
diagonal  plane. 


A  polyedral  is  called  convex,  when  every  diagonal  plane 
lies  within  the  figure  ;  otherwise  it  is  called  concave. 

When  any  figure  is  cut  by  a  plane,  the  figure  that  is  defined 
on  the  plane  by  the  surfaces  of  the  figure  so  cut  is  called  a 
plane  section. 

Corollaries, — I.  If  the  plane  of  one  face  of  a  convex  polye- 
dral is  produced,  it  does  not  cut  the  polyedral. 

II.  A  plane  may  pass  through  the  vertex  of  a  convex  polye- 
dral without  cutting  any  face. 


Art.  40.]  STRAIGHT  LINES  AND  PLANES.  183 

III.  A  plane  may  cut  all  the  edges  of  a  convex  polyedral. 
The  section  is  a  convex  polygon. 


41.  Theorem. — The  sum  of  the  faces  of  a  convex  polyedral 
is  less  than  four  right  angles. 

This  is  demonstrated  in  the  same  manner  as  the  correspond- 
ing theorem  on  triedrals  (32). 

42.  Theorem. — Tn  any  convex  polyedral^  the  sum  of  the 
diedrals  is  greater  than  the  sum  of  the  angles  of  a  polygon 
haviyig  the  same  number  of  sides  that  the  polyedral  has 
faces. 

Let  the  given  polyedral  be  divided  by  diagonal  planes  into 
triedrals.  Then  this  theorem  is  demonstrated  like  the  analogous 
proposition  on  polygons. 

43.  Scholium.. — The  theory  of  equality  of  triedrals  has 
some  analogy  to  the  theory  of  similarity,  and  some  to  the  theory 
of  equality  of  triangles.  In  the  discussion  of  triedrals  there  is 
no  consideration  of  the  ratios  of  lengths  as  in  similarity,  nor  of 
the  equality  of  magnitudes  as  in  the  equality  of  triangles. 
Every  element  of  a  triedral  is  an  angle.  The  analogy  of  trie- 
drals with  triangles  will  be  more  apparent  in  the  theoiy  of 
spherical  triangles,  which  is  based  upon  the  theory  of  trie- 
drals. 

At  first  it  appears  impossible  to  adapt  problems  in  draw- 
ing to  the  Geometry  of  Space  ;  for  a  drawing  is  made  on 
a  plane  surface,  while  the  figures  investigated  are  not  plane 
figures. 

This,  however,  has  been  accomplished  by  the  most  ingenious 
methods,  invented,  in  great  part,  by  Monge,  one  of  the  founders 
of  the  Polytechnic  School  at  Paris,  the  first  who  reduced  to 
a  system  the  elements  of  this  science,  called  Descriptive  Geome- 
try. 

Descriptive  Geometry  is  that  branch  of  mathematics  which 
teaches  how  to  represent  and  determine,  by  means  of  drawings 
on  a  plane  surface,  the  absolute  or  relative  position  of  points  or 


184  GEOMETRY  OF  SPACE.  [Chap.  IX. 

magnitudes  in  space.  It  is  beyond  the  design  of  the  present 
work  to  do  more  than  allude  to  this  interesting  and  very  useful 
science. 


44.  Exercises. — 1.  If  a  straight  line  is  perpendicular  to  a 
plane,  every  plane  parallel  to  the  given  line  is  perpendicular  to  the 
given  plane. 

2.  If  every  diedral  of  a  triedral  is  bisected,  the  three  planes  have  one 
common  intersection. 

3.  If  two  straight  lines  are  not  in  the  same  plane,  there  is  one  straight 
line  and  only  one  that  is  perpendicular  to  both  of  them. 

4.  In  the  second  case  of  Exercise  5,  Article  28,  a  line  that  is  per- 
pendicular to  both  the  given  lines  is  also  perpendicular  to  the  two 
planes. 

5.  When  two  parallel  planes  are  cut  by  a  third  plane,  the  corre- 
sponding diedrals  are  equal.     Is  the  converse  true  ? 

6.  When  two  triedrals  have  two  faces  of  the  one  respectively  equal  to 
two  faces  of  the  other,  and  the  included  diedrals  unequal,  the  third  faces 
are  unequal,  and  that  face  is  greater  which  is  opposite  the  greater  die- 
dral.    Is  the  converse  true  ? 

T.  If  two  convex  polyedrals  have  the  same  number  of  faces,  the  sum 
of  the  diedrals  of  one  is  within  four  right  angles  of  the  sum  of  the  die- 
drals of  the  other. 


CHAPTER  X. 
POLYEDRONS. 

Article  1. — Figures  that  inclose  a  portion  of  space  are 
classified  according  to  their  surfaces.  This  chapter  treats  of 
some  that  have  plane  surfaces  ;  the  following,  of  some  that  have 
curved  surfaces. 

A  Polyedron  is  a  solid,  or  portion  of  space,  bounded  by 
plane  surfaces.  Each  of  the  surfaces  is  a  face,  their  intersec- 
tions are  edges,  and  the  points  of  meeting  of  the  edges  are  ver- 
tices of  the  polyedron. 

Corollary. — The  edges  of  a  polyedron  are  straight  lines  (IX, 
4),  and  the  faces  are  polygons. 

A  Diagonal  of  a  polyedron  is  a  straight  line  joining  two 
vertices  that  are  not  in  the  same  face. 

A  Diagonal  Plane  is  a  plane  passing  through  three  ver- 
tices that  are  not  in  the  same  face. 

Tetraedrons. 

2.  A  Tetraedron  is  a  polyedron  having  four  faces. 

Three  planes  can  not  inclose  a  solid  (IX,  29),  but  if  a  plane 
is  passed  through  three  points,  one  on  each  edge  of  a  triedral, 
the  triangular  plane  section  and  the  three  triangles  cut  off  in- 
close a  polyedron,  which  has  four  faces,  four  vertices  and  six 
edges. 

Problem. — A7iy  four  points,  not  all  in  one  plane,  may  he 
the  vertices  of  a  tetraedron. 

Straight  lines  joining  these,  two  and  two,  form  the  six  edges, 
and  these  bound  the  four  faces. 
9 


186  *  GEOMETRY  OF  SPACE  [Chap.  X. 

The  Altitude  of  a  tetraedron  is  the  perpendicular  distance 
from  one  face  to  the  opposite  vertex.  This  face  is  called  the 
base,  and  the  vertex  is  called  the  vertex  of  the  tetraedron. 

Any  face  of  a  tetraedron  may  be  taken  as  the  base  ;  then  the 
other  faces  are  called  the  sides.  Consequently  the  altitude  may 
be  either  of  four  distances.  When  a  diedral  edge  of  the  base  is 
obtuse,  the  perpendicular  falls  outside  of  the  figure,  on  the  base 
produced. 

Corollary. — ^The  altitude  of  a  tetraedron  is  equal  to  the  dis- 
tance from  the  base  to  a  plane  through  the  vertex  parallel  to  the 
base  (IX,  23). 

3.  Theorem. — There  is  one  point  equally  distant  from  the 
four  vertices  of  a  tetraedron. 

Every  point  that  is  equally  distant  from  the  three  points  B, 
C,  and  jP]  lies  in  the  axis  of  the  circle  whose  circumference 
passes  through  those  three 
points  (IX,  8,  iv).  Every  point 
equally  distant  from  the  two 
points  G  and  D  lies  in  the 
plane  that  bisects  the  edge 
CD    perpendicularly    (IX,    6, 

V). 

Since  the  line  CD  is  not 
parallel  to  the  plane  B  CF,  the 
plane  perpendicular  to  CD  and 
the  line  perpendicular  to  B  CF 

are  not  parallel,  and  the  plane  must  cut  the  line  at  some  point. 
The  point  common  to  these  two  loci  is  the  only  point  equally 
distant  from  the  four  vertices. 

The  point  is  not  necessarily  within  the  tetraedron. 

Corollaries. — I.  The  axes  of  the  four  circles  that  circum- 
scribe the  faces  of  a  tetraedron  meet  at  one  point. 

II.  The  six  planes  that  bisect  perpendicularly  the  edges  of  a 
tetraedron  meet  at  the  same  point. 

4.  Theorem. — There  is  one  point  xoithin  every  tetraedron 
equally  distant  from  the  four  faces. 


Art.  4.]  P0LYEDR0N8.  187 

Let  OB  be  the  strai^t  line  formed  by  the  intersection  of 
two  planes,  one  of  which  bisects  the  diedral  A  0,  and  the  other 
the  diedral  EO. 

Every  point  of  the  first  ^ 

bisecting    plane    is    equally  ^-^</^^ 

distant  from  the  faces  lA  O  ^^/       \ 

and  EA  0  ;  and  every  point  ^^    /  \ 

of  the  second  bisecting  plane         JE*^- ----/-  ---^--rn^^  0 

is  equally  distant  from  the  ^^/L^""^ — 

faces  ^^0  and  ^JO.  There-  i 

fore,  every  point  of  the  line 

B  0  is  equally  distant  from  those  three  faces. 

Then  let  a  plane  bisect  the  diedral  EI,  and  let  C  be  the  point 
where  this  plane  cuts  the  line  B  0. 

Since  every  point  of  this  last  bisecting  plane  is  equally  dis- 
tant from  the  faces  EAI  and  EOI,  the  point  C  is  equally  dis- 
tant from  the  four  faces  of  the  tetraedron. 

Since  all  the  bisecting  planes  are  interior,  the  point  found  is 
within  the  tetraedron. 

Corollary. — The  six  planes  that  bisect  the  diedrals  of  a  tetra- 
edron all  meet  at  one  point. 


Pyramids. 

5,  A  Pyramid  is  a  polyedron  having  for  one  face  a  poly- 
gon, and  for  the  other  faces  triangles  whose  vertices  are  at  one 
point. 

"When  a  plane  cuts  all  the  edges  of  a  polyedral,  the  portion 
of  space  cut  off  is  a  pyramid. 

The  polygon  is  the  base 
of  the  pyramid,  the  triangles 
are  its  sides,  and  their  in- 
tersections are  the  lateral 
edges  of  the  pyramid.  The 
vertex  of  the  polyedral  is 
the  vertex  of  the  pyramid, 
and  the   perpendicular  dis- 


188  GEOMETRY  OF  SPACE.  [Chap.  X. 

tance  from  that  point  to  the  plane  of  the  base  is  its  alti- 
tude. 

Pyramids  are  called  triangular,  quadrangular,  pentagonal, 
etc.,  according  to  the  polygon  which  forms  the  base.  The  tet- 
raedron  is  a  triangular  pyramid. 

A  Regular  Pyramid  is  one  whose  base  is  a  regular  poly- 
gon, and  whose  vertex  is  in  the  line  perpendicular  to  the  base  at 
its  center. 

The  Slant  Height  of  a  regular  pyramid  is  the  perpendicular 
from  the  vertex  upon  one  side  of  the  base.  It  is  therefore  the 
altitude  of  one  of  the  sides  of  the  pyramid. 

When  a  pyramid  is  cut  by  a  plane  parallel  to  the  base,  that 
part  of  the  figure  between  this  plane  and  the  base  is  called  a 
frustum  of  a  pyramid,  or  a  truncated  pyramid. 

The  section  made  by  the  cutting  plane  is  called  the  upper 
base  of  the  frustum.  The  slant  height  of  the  frustum  of  a 
regular  pyramid  is  that  part  of  the  slant  height  of  the  pyramid 
which  lies  between  the  bases  of  the  frustum.  It  is  therefore  the 
altitude  of  one  of  the  sides. 

Corollaries. — I.  The  lateral  edges  of  a  regular  pyramid  are 
equal,  and  the  sides  are  equal  isosceles  triangles. 

11.  The  sides  of  a  frustum  of  a  pyramid  are  trapezoids.  The 
sides  of  the  frustum  of  a  regular  pyramid  are  equal,  and  they 
are  symmetrical  figures. 

Problem. — A  pyramid  can  he  divided  into  the  same  number 
of  tetraedrons  as  its  base  can  he  divided  into  triangles. 

Let  a  diagonal  plane  pass  through  the  vertex  of  the  pyramid 
and  each  diagonal  of  the  base. 

Corollary. — Any  polyedron  can  be  divided  into  tetraedrons. 
For  a  concave  polyedron  can  be  divided  into  convex  polyedrons 
by  diagonal  planes,  and  every  convex  polyedron  can  be  divided 
into  pyramids  having  the  several  faces  for  their  bases  and  a 
common  point  for  vertex. 


Art.  6.] 


POLYEDRONS. 


189 


Prisms. 

6.  A  Prism  is  a  polyedron  which  has  two  of  its  faces 
equal  polygons  lying  in  parallel  planes,  and  the  other  faces 
parallelograms.  Its  possibil- 
ity is  shown  by  supposing 
two  polygons  with  sides  re- 
spectively equal  and  paral- 
lel, lying  in  parallel  planes ; 
the  equal  sides  being  paral- 
lel, let  planes  unite  them. 
The  plane  figures  thus  formed 
are  parallelograms,  for  every 
one  has  two  opposite  sides 
equal  and  parallel. 

The  parallel  polygons  are  called  the  bases,  the  parallelograms 
the  sides  of  the  prism,  and  the  intersections  of  the  sides  are  its 
lateral  edges. 

The  Altitude  of  a  prism  is  the  perpendicular  distance  be- 
tween the  planes  of  its  bases. 

A  Right  Prism  is  one  whose  lateral  edges  are  perpendicu- 
lar to  the  bases. 

A  Regular  Prism  is  a  right  prism  whose  base  is  a  regular 
polygon. 

A  Parallelepiped  is  a 
prism  whose  bases  are  parallel- 
ograms. Hence,  a  parallelo- 
piped  is  a  solid  inclosed  by  six 
parallelograms. 

When  the  bases  of  a  right 
parallelopiped  are  rectangles, 
it  is  called  rectangular.  De 
Morgan  proposed  for  this  figure 
the  more  simple  term  right  solid.     Every  angle  in  it  is  right. 

A  Cube  is  a  rectangular  parallelopiped  whose  length, 
breadth  and  altitude  are  equal.  Then  a  cube  is  bounded  by  six 
equal  squares  ;  its  vertices  are  trirectangular  triedrals ;  and  its 
edges  are  of  right  diedral  angles. 


190  GEOMETRY  OF  SPACE.  [Chap.  X. 

Corollaries. — I.  The  lateral  edges  of  a  prism  are  parallel, 
and  equal. 

II.  The  altitude  of  a  right  prism  is  equal  to  one  of  its  lateral 
edges  ;  and  the  sides  of  a  right  prism  are  rectangles.  The  sides 
of  a  regular  prism  are  equal. 

III.  Two  opposite  faces  of  a  parallelopiped  are  equal. 

IV.  Any  two  opposite  faces  of  a  parallelopiped  may  be  as- 
sumed as  the  bases  of  the  figure. 


7.  Theorem. — If  tioo  parallel  planes  cut  all  the  lateral  edges 
of  a  prisnij  that  part  of  the  solid  between  these  planes  is  a 
prism. 

Every  side  of  one  section  is  parallel  to  the  corresponding  side 
of  the  other  section  (IX,  21).  Hence,  that  portion  of  each  side 
of  the  prism  which  is  between  the  secant  planes  is  a  parallelo- 
gram. Since  the  sections  have  their  sides  respectively  equal  and 
parallel,  their  angles  are  respectively  equal.  Therefore,  the 
polygons  are  equal. 

Corollary. — A  section  of  a  prism  by  a  plane  parallel  to  the 
base  is  equal  to  the  base,  and  the  prism  is  divided  into  two 
prisms. 

8.  Problem, — A  prism  can  he  divided  into  the  same  num- 
ber of  triangular  prisms  as  its  base  can  be  divided  into  trian- 
gles. 

Homologous  diagonals  made  in 
the  two  bases,  as  JS'O  and  CI]  lie  in 
one  plane.  For  CU  and  OJF^  being 
parallel,  lie  in  one  plane.  There- 
fore, through  every  pair  of  homolo- 
gous diagonals  a  plane  may  pass. 
These  diagonal  planes  divide  the 
prism  into  triangular  prisms. 


9,  Problem, — A  triangular  prism  can  be  divided  into  three 
tetraedrons. 


Art.  9.]  P0LYEDR0N8.  191 

A  diagonal  plane  may  pass  through  the  points  J5,  (7,  and  H, 
making  the  intersections  JSII  and  CHj  in  the  sides  J)F  and 
DCr.  This  plane  cuts  off  the  tet- 
raedron  BGDH,  which  has  for  one 
of  its  faces  the  base  BCD  of  the 
prism  ;  for  a  second  face,  the  triangle 
B  CH,  being  the  section  made  by  the 
diagonal  plane  ;  and  for  its  other  two 
faces,  the  triangles  BBS  and  CDH, 
each  being  half  of  one  of  the  sides  of 
the  prism. 

The  remainder  of  the  prism  is  a 
quadrangular    pyramid,    having     the 

parallelogram  BGGF  for  its  base,  and  H  for  its  vertex.  A 
diagonal  plane  may  pass  through  the  points  H,  6r,  and  J?, 
dividing  this  pyramid  into  two  tetraedrons,  IIBGG-  and 
HBFG, 

The  faces  HBG  and  HBG,  of  the  tetraedron  HBGG,  are 
sections  made  by  the  diagonal  planes  ;  and  the  faces  HGG  and 
BGG  are  halves  of  sides  of  the  prism.  The  tetraedron  HBFG 
has  for  one  of  its  faces  the  base  HFG  of  the  prism  ;  for  a  second 
face,  the  triangle  HBG,  being  the  section  made  by  the  diagonal 
plane  ;  and,  for  the  other  two,  the  triangles  HBF  and  GBF, 
halves  of  sides  of  the  prism. 

10,  Theorem. — TTie  three  tetraedrons  which  compose  a  tri- 
angular prism,  taken  two  and  two,  have  equal  bases  and  equal 
altitudes. 

Consider  the  last  two  tetraedrons  described  in  the  previous 
article  as  having  their  bases  BGG  and  BFG.  These  are  equal 
triangles  lying  in  one  plane.  The  point  ^is  the  common  vertex 
of  the  tetraedrons.  Therefore  they  have  the  same  altitude  ; 
that  is,  a  perpendicular  from  ^to  the  plane  BGGF. 

Next,  consider  the  first  and  last  tetraedrons  described,  HB  GD 
and  BFGH,  the  former  as  having  B  GD  for  its  base,  and  H  for 
its  vertex  ;  the  latter  as  having  FGH  for  its  base,  and  B  for  its 
vertex.  These  bases  are  equal,  being  the  bases  of  the  given 
prism.     The  vertex  of  each  is  in  the  plane  of  the  base  of  the 


192 


GEOMETRY  OF  SPACE. 


[Chap.  X. 


Other.     Therefore,  the   altitudes  are  equal,  being  the  distance 
between  these  two  planes. 

Lastly,  consider  the  tetraedrons 
BCDH  and  BCGH  as  having  their 
bases  CDH  and  CGH,  These  are 
equal  triangles  lying  in  one  plane. 
The  tetraedrons  have  the  common 
vertex  B,  and  hence  have  the  same 
altitude. 

Scholium. — On  account  of  the  im- 
portance of  the  above  problem  in 
future  demonstrations,  the  student  is 

advised  to  make  a  model  triangular  prism,  and  divide  it  into 
tetraedrons.     A  potato  may  be  used  for  this  purpose. 


Regular  Polyedrons. 

11.  A  Regular  Polyedron  is  one  whose  faces  are  equal 
and  regular  polygons,  and  whose  vertices  are  equal  polye- 
drals. 

Thus  the  faces  of  a  regular  tetraedron  are  equal  equilateral 
triangles,  and  the  vertices  are  of  equal  triedrals. 

The  regular  hexaedron  is  a  regular  quadrangular  prism, 
whose  six  faces  are  equal  squares,  and  whose  right  vertices  are 
of  trirectangular  triedrals.     This  is  a  cube. 


The  regular  octaedron  has  eight,  the  dodecaedron  twelve, 
and  the  icosaedron  twenty  faces. 


Art.  11.]  P0LYEDR0N8.  193 

The  class  of  figures  here  defined  must  not  be  confounded 
with  regular  pyramids  or  prisms. 

Problem. —  Only  five  regular  polyedrons  are  possible. 

First,  consider  those  whose  faces  are  triangles.  Each  angle 
of  a  regular  triangle  is  one  third  of  two  right  angles.  Either 
three,  four,  or  five  of  these  may  be  joined  to  form  one  polyedral 
vertex,  the  sum  being,  in  each  case,  less  than  four  right  angles. 
But  the  sum  of  six  such  angles  is  not  less  than  four  right  angles. 
Therefore,  there  can  not  be  more  than  three  kinds  of  regular 
polyedrons  whose  faces  are  triangles,  viz.  :  the  tetraedron, 
where  three  plane  angles  form  a  vertex ;  the  octaedron, 
where  four,  and  the  icosaedron,  where  five  angles  form  a 
vertex. 

The  same  kind  of  reasoning  shows  that  only  one  regular 
polyedron  is  possible  with  square  faces,  the  cube  ;  and  only  one 
with  pentagonal  faces,  the  dodecaedron  ;  and  that  the  faces  of  a 
regular  polyedron  can  not  be  polygons  of  six  or  more  sides. 

The  construction  of  the  regular  tetraedron  (2)  and  of  the 
regular  hexaedron  (6)  has  been  explained. 

For  the  construction  of  the  octaedron  there  may  be  three 
equal  straight  lines,  two  of  them  bisecting  each  other  perpendic- 
ularly, and  the  third  perpendicular  to  the  plane  of  those  two 
and  bisected  by  it.  Joining  the  extremities  by  twelve  straight 
lines,  these  lines  are  equal  (V,  11),  and  are  the  sides  of  eight 
equal  equilateral  triangles,  which  are  the  faces  of  a  regular  oc- 
taedron. 

The  possibility  of  regular  polyedrons  of  twelve  and  of  twenty 
faces  is  here  assumed,  as  the  solution  is  tedious  and  of  little  value. 
The  student  may  construct  models  by  the  plans  given  in  Article 
17. 

Corollaries. — I.  Reasoning  as  in  the  case  of  a  tetraedron, 
there  is  a  point  in  every  regular  polyedron  equally  distant  from 
all  the  vertices  (3).  This  point,  called  the  center,  is  equally  dis- 
tant from  all  the  faces,  and  it  is  equally  distant  from  all  the 
edges. 

II.  Every  regular  polyedron  may  be  divided  into  as  many 
equal  regular  pyramids  as  the  polyedron  has  faces. 

III.  The  centers  of  the  faces  of  a  cube  may  be  the  vertices 


194  GEOMETRY  OF  SPACE.  [Chap.  X. 

of  a  regular  octaedron  ;  and  the  centers  of  the  faces  of  a  regular 
octaedron  may  be  the  vertices  of  a  cube. 


13.  Exercises. — 1.  Should  it  be  stated  as  a  condition  of  the 
problem  in  Article  2  that  no  three  of  the  points  are  in  one  straight 
line? 

2.  In  a  tetraedron,  the  three  lines  that  join  the  midpoints  of  opposite 
edges  bisect  each  other. 

3.  If  one  of  the  vertices  of  a  tetraedron  is  a  trirectangular  triedral,  the 
square  of  the  area  of  the  opposite  face  is  equal  to  the  sum  of  the  squares 
of  the  areas  of  the  other  three  faces. 

4.  A  pyramid  is  regular  when  its  sides  are  equal  isosceles  triangles 
whose  bases  form  the  perimeter  of  the  base  of  the  pyramid. 

5.  How  many  edges  and  how  many  vertices  has  each  of  the  regular 
polyedrons  ? 

6.  The  three  tetraedrons  which  compose  a  triangular  prism,  taken  two 
and  two,  have  two  faces  of  one  respectively  equal  to  two  faces  of  the 
other;  and  taking  the  equal  faces  as  bases  the  corresponding  altitudes  are 
equal ;  that  is,  every  conclusion  of  the  tlieorem,  Article  10,  can  be  other- 
wise demonstrated. 

7.  Every  regular  pyramid  is  symmetrical.  How  many  planes  of 
symmetry  has  it?  When  has  it  an  axis  of  symmetry?  Has  it  a  center 
of  symmetry  ? 

8.  Every  regular  prism  is  symmetrical.  Has  it  planes?  an  axis?  a 
center  of  symmetry  ? 

9.  Has  every  parallelepiped  axes  of  symmetry?  Planes?  Has  it  a 
center  of  symmetry  ? 

Equality  of  Polyedrons. 

13.  Theorem. — Equal  polyedrons  are  composed  of  tetrae- 
drons respectively  equal  and  similarly  arranged;  and  con- 
versely. 

These  principles  are  corollaries  of  the  definition  of  equal- 
ity. 

When  two  polyedrons  are  symmetrical  to  each  other,  each 
line,  angle  or  surface  in  one  is  equal  to  the  corresponding  ele- 
ment in  the  other ;  but  the  elements  are  arranged  in  reverse 
order.  Conversely,  two  polyedrons  so  composed  are  symmetri- 
cal to  each  other. 


Art.  14.]  P0LYEDR0N8.  195 

14.  Theorem. — Two  tetraedrons  are  equal  when  the  six 
edges  of  one  are  respectively  equal  to  those  of  the  other,  and  are 
similarly  arranged. 

For  the  corresponding  faces  are  equal  (V,  10),  and  the  trie- 
dral  vertices  {IX,  35).  In  a  word,  every  element  of  one  figure 
is  equal  to  the  corresponding  element  of  the  other. 

Corollaries. — The  elements  being  similarly  arranged,  two 
tetraedrons  are  equal : 

I.  When  three  faces  of  one  are  respectively  equal  to  three 
faces  of  the  other  ; 

II.  When  two  faces  and  the  included  diedral  of  one 
are  respectively  equal  to  those  elements  of  the  other  (IX, 
37); 

III.  When  one  face  and  the  adjacent  diedrals  of  one  are 
respectively  equal  to  those  elements  of  the  other  (IX,  38). 

15.  Theorem. — Two  pyramids  are  equal  when  they  have 
equal  bases,  and  three  lateral  edges  of  one  respectively  equal  to 
those  of  the  other  and  similarly  arranged. 

Since  the  bases  are  equal,  their  diagonals  and  sides  are  re- 
spectively equal.  The  three  lines  in  each  base  joining  the  ver- 
tices corresponding  to  the  given  lateral  edges,  and  these  three 
edges,  constitute  six  edges  of  a  tetraedron,  whose  vertex  is  the 
vertex  of  the  pyramid.  Therefore,  if  the  one  pyramid  is  placed 
on  the  other  so  that  the  equal  bases  coincide,  the  vertices  must 
coincide.  Consequently  all  the  lateral  edges  coincide,  and  the 
figures  coincide  throughout. 

Corollary. — Two  regular  pyramids  are  equal  if  they  have 
equal  bases  and  equal  altitudes. 

16.  Theorem. —  Two  prisms  are  equal  when  a  base,  a  trie- 
dral  vertex,  and  a  lateral  edge  of  one  are  respectively  equal  to 
the  corresponding  elements  of  the  other  and  are  similarly  ar- 
ranged. 

The  demonstration  is  left  to  the  student. 
Corollary. — Two  right  prisms   are  equal  when  they  have 
equal  bases  and  the  same  altitude. 


196 


GEOMETRY  OF  SPACR 


[Chap.  X. 


17.  Scholium. — Paper  models  of  some  of  the  polyedrons 
which  have  been  defined  may  be  constructed  as  follows  : 

First,  the  tetraedron,  the  six  edges  being  given  ; 

With  three  of  the  edges  that  are  sides  of  one  face,  draw  the 
triangle,  as  AB  C,     On  each  side  of  this  triangle,  as  a  base,  draw 
a  triangle  equal  to  the 
corresponding    face;  /^i> 

which  can  be  done, 
for  the  sides  of  these 
triangles  are  given. 
Then,  cut  out  the 
whole  figure  from  the 
paper  and  fold  it  at 
the  lines  AB^  B  (7,  and 

CA.     Since  BF  is  equal  to  BI),  CF  to  GE,  and  AB  to  AE, 
the  points  F,  I),  and  E  may  be  united  to  form  a  vertex. 

To  construct  models  of  symmetrical  tetraedrons  make  the 
drawings  equal,  but  fold  up  in  one  and  down  in  the  other. 

A  regular  pyramid  :  Draw  a  regular  polygon  of  any  number 
of  sides.  Upon  these  sides,  as  bases,  draw  equal  isosceles  trian- 
gles, with  altitudes  greater  than  the  apothem  of  the  base.  Cut 
out  and  fold. 

Right  prisms  present  no  difficulty.  Other  prisms  and  pyramids 
and  frustra  may  be  constructed  by  trial  with  tolerable  accuracy. 

The  five  regular  polyedrons  :  For  the  regular  tetraedron, 
draw  an  equilateral  triangle,  join  the  midpoints  of  the  three 
sides,  cut,  and  fold. 


For  the  cube,  draw  six  equal  squares,  as  in  the  diagram. 
For  the  octaedron,  draw  eight  equal  regular  triangles. 


Art.  n.]  P0LYEDR0N8.  197 

For  the  dodecaedron,  draw  twelve  equal    regular    penta- 
gons. 


For  the  icosaedron,  draw  twenty  equal  regular  triangles. 

18,  Exercises. — 1.  Are  two  tetraedrons  equal  when  five  edges  of 
one  are  respectively  equal  to  five  edges  of  the  other,  the  edges  being 
similarly  arranged  ? 

2.  Are  two  regular  hexagonal  pyramids  equal,  if  they  have  equal  alti- 
tudes and  a  side  cf  one  hase  equal  to  a  side  of  the  other  ? 

3.  If  two  triangular  prisms  are  composed  of  tetraedrons  respectively 
equal,  must  the  prisms  be  equal  ? 

4.  If  a  right  prism  is  symmetrical  to  another,  they  are  equal. 

5.  Two  regular  pyramids,  symmetrical  to  each  other,  are  equal;  and 
conversely. 

Similarity  of  Polyedrons. 

1 9.  Since  similarity  consists  in  having  the  same  form,  so 
that  every  difference  of  direction  in  one  of  two  similar  figures 
has  its  corresponding  equal  difference  of  direction  in  the  other, 
it  follows  that,  when  two  polyedrons  are  similar,  their  homolo- 
gous faces  are  similar  polygons,  their  homologous  edges  are  of 
equal  diedral  angles,  and  their  homologous  vertices  are  of  equal 
polyedrals. 

If  the  similar  faces  are  not  arranged  similarly,  but  in  reverse 
order,  the  polyedrons  are  symmetrically  similar. 

Theorem. —  When  two  polyedrons  are  similar,  any  edge  or 
other  line  in  one  is  to  the  homologous  line  in  the  second  as  any 
other  line  in  the  first  is  to  its  homologous  line  in  the  second. 

If  the  proportion  to  be  proved  is  between  sides  of  homolo- 


398  GEOMETRY  OF  SPACE.  [Chap.  X. 

gous  triangles,  it  follows  at  once  from  the  similarity  of  the  tri- 
angles. 

Suppose  the  edges  taken  in  one  of  the  polyedrons  are  not 
sides  of  one  face  ;  as,  AE  and  10,  Then  it  is  to  be  proved  that 
AE:BG  =  IO:DF. 

AE '.BC  =  IE'.CD,2i%  just  proved, 
and  IO'.DF=zIE:CD. 

Therefore,  AE\BC  =  IO\  DF, 

Again,  compare  the  homologous  altitudes  AK  and  BH. 
Join  KO  and  HF.  Then  the  planes  KA  O  and  HBF  are  re- 
spectively perpendicular  to  the  bases  EIO  and  CDF,  and  the 


angles  KOA  and  JIFB  are  homologous  and  equal.     Then  the 
right-angled  triangles  KOA  and  HFB  are  similar,  and 

AKiBR=AO:BF. 

Thus  any  homologous  lines  can  be  united  to  homologous 
edges  by  similar  triangles. 

Corollary, — When  two  polyedrons  are  symmetrically  similar 
homologous  lines  are  proportional. 

20.  Theorem. —  Two  tetraedrons  are  similar  when  there  is 
the  same  ratio  of  every  edge  of  one  to  the  corresponding  edge  of 
the  other,  and  they  are  similarly  arranged. 

Since  the  corresponding  faces  are  similar  triangles,  every 
angle  on  the  surface  of  one  has  its  corresponding  equal  angle  on 
the  surface  of  the  other. 

If  a  line  is  made  through  the  figure,  it  is  shown,  by  the  aid  of 
auxiliary  lines,  as  in  the  corresponding  proposition  of  similar  tri- 


Art.  20.]  POLYEDRONS.  199 

angles,  that  every  angle  in  one  figure  has  its  homologous  equal 
angle  in  the  other. 

Corollaries. — The  corresponding  elements  being  similarly- 
arranged,  two  tetraedrons  are  similar  : 

I.  When  three  faces  of  one  are  respectively  similar  to  those 
of  the  other ; 

II.  When  two  triedral  vertices  of  one  are  respectively  equal 
to  two  vertices  of  the  other. 

III.  If  a  plane  cuts  a  tetraedron  parallel  to  one  face,  the  part 
cut  off  is  similar  to  the  whole. 

21.  Theorem. — Two  similar  polyedrons  are  composed  of 
tetraedrons  respectively  similar  and  similarly  arranged. 

For,  however  the  one  is  divided  into  tetraedrons,  the  con- 
struction of  homologous  lines  and  planes  divides  the  other  in  the 
same  manner.  Then  the  similarity  of  the  corresponding  tetrae- 
drons follows  from  the  proportionality  of  the  lines. 

Corollaries. — I.  Conversely,  two  polyedrons  are  similar 
when  composed  of  tetraedrons  respectively  similar  and  similarly 
arranged. 

II.  When  two  polyedrons  are  similar  there  is  the  same  ratio 
of  any  line  of  one  to  the  homologous  line  of  the  other. 

III.  When  a  pyramid  is  cut  by  a  plane  parallel  to  the  base, 
the  pyramid  cut  off  is  similar  to  the  whole. 

23,  Theorem. — If  lines  are  tnade  from  the  vertices  of  a 
polyedron  to  any  point  in  space,  and  if  these  lines  are  divided 
either  internally  or  externally  in  the  same  ratio,  the  points  of 
division  are  the  vertices  of  a  polyedron,  which  is  either  similar 
or  symmetrically  similar  to  the  first. 

Conversely,  if  two  similar  or  symmetrically  similar  polye- 
drons have  their  homologous  edges  parallel,  lines  made  through 
homologous  points  all  meet  in  one  point. 

The  demonstration  of  these  propositions  is  similar  to  that  of 
the  corresponding  propositions  in  Plane  Geometry  (V,  27  and 
28).     The  detail  of  the  argument  is  left  to  the  student. 

The  definition  of  internal  and  of  external  centers  of  simi- 
larity is  as  in  Plane  Geometry. 


200  GEOMETRY  OF  SPACE.  [Chap.  X. 

Scholium. — The  contrary  arrangement  of  the  parts  in  plane 
figures  arises  from  viewing  the  figures  from  an  opposite  direc- 
tion. The  arrangement  of  the  parts  viewed  from  the  reverse 
face  is  contrary  to  that  which  they  have  when  viewed  from  the 
obverse  face  of  the  figure.  Change  in  the  arrangement  of  the 
parts  of  a  solid  figure  can  not  be  effected  by  any  change  of  the 
position  of  the  observer ;  the  contrary  arrangement  makes  a 
different  figure. 


Surfaces  of  Polyedrons. 

23.  The  area  of  the  surface  of  a  polyedron  is  the  sum  of  the 
areas  of  the  faces.  When  several  faces  are  equal,  the  process  is 
shortened  by  multiplication. 

Theorem. — The  area  of  the  lateral  surface  of  a  regular 
pyramid  is  equal  to  half  the  product  of  the  perirneter  of  the  base 
by  the  slant  height. 

The  area  of  every  side  is  equal  to  half  the  product  of  its  base 
by  its  altitude.  But  the  altitude  of  every  side  is  the  slant  height 
of  the  pyramid,  and  the  sum  of  the  bases  of  the  sides  is  the 
perimeter  of  the  base  of  the  pyramid. 

Therefore,  the  area  of  the  lateral  surface  of  the  pyramid, 
which  is  the  sum  of  all  the  sides,  is  equal  to  half  the  product  of 
the  perimeter  of  the  base  by  the  slant  height. 

Corollary. — The  area  of  the  lateral  surface  of  a  regular  pyra- 
mid is  equal  to  the  product  of  the  slant  height  by  the  perimeter 
of  a  section,  midway  between  the  vertex  and  the  base.  For  the 
perimeter  of  the  midway  section  is  one  half  the  perimeter  of  the 
base. 


34.  Theorem, — The  area  of  the  lateral  surface  of  the  frus- 
tum of  a  regular  pyramid  is  equal  to  half  the  product  of  the 
sum  of  the  perimeters  of  the  bases  by  the  slant  height. 

The  area  of  every  trapezoidal  side  is  equal  to  half  the  prod- 
uct of  the  sum  of  its  parallel  bases  by  its  altitude,  which  is  the 
slant  height  of  the  frustum.  Therefore,  the  area  of  the  lateral 
surface,  which  is  the  sum  of  these  equal  trapezoids,  is  equal  to 


Abt.  24.] 


P0LYEDR0N8. 


201 


the  product  of  half  the  sum  of  the  perimeters  of  the  bases  multi- 
plied by  the  slant  height. 

Corollary, — ^The  area  of  the  lateral  surface  of  a  frustum  of  a 
regular  pyramid  is  equal  to  the  product  of  the  slant  height  by 
the  perimeter  of  a  section  midway  between  the  bases. 


25.  Theorem. — The  area  of  the  lateral  surface  of  a  prism 
is  equal  to  the  product  of  one  of  the  lateral  edges  by  the  perim- 
eter of  a  section^  made  hy  a  plane  perpendicular  to  those 
edges. 

Since  the  plane  UN'  is  perpendicular  to  the  edges  of  the 
prism,  every  side  of  the  polygon  HJST  is 
perpendicular  to  the  edges  which  it  con- 
nects. Every  side  of  the  prism  is  a  par- 
allelogram with  one  lateral  edge  of  the 
prism  for  its  base  and  one  side  of  the  poly- 
gon IIN"  for  its  altitude  ;  and  the  area  of 
the  parallelogram  is  the  product  of  these 
two  factors. 

Now,  the  lateral  edges  of  the  prism  are 
equal.     Therefore,  the  sum  of  the  areas  of 
these  parallelograms,  that  is,  the   lateral 
surface  of  the  prism,  is  equal  to  the  product  of  one  edge  multi- 
plied by  the  perimeter  of  the  polygon. 

Corollary. — The  area  of  the  lateral  surface  of  a  right  prism 
is  equal  to  the  product  of  the  altitude  by  the  perimeter  of  the 
base. 


26.  Theorem.— The  areas  of  homologous  faces  of  similar 
tetraedrons  are  to  each  other  as  the  squares  of  homologous 
lines. 

This  is  a  corollary  of  the  theorem  that  the  areas  of  simi- 
lar triangles  are  to  each  other  as  the  squares  of  homologous 
lines. 

Corollary. — The  areas  of  homologous  surfaces  of  similar 
polyedrons  are  to  each  other  as  the  squares  of  homologous 
lines. 


202  GEOMETRY  OF  SPACK  [Chap.  X. 

27.  Theorem. — If  two  tetraedrons,  having  the  same  altitude 
and  their  bases  on  the  same  plane,  are  cut  by  a  plane  parallel  to 
their  bases,  the  areas  of  the  sections  have  the  same  ratio  as  the 
areas  of  the  bases. 

The  frustra  have  equal  altitudes.  Subtracting  these  from  the 
given  common  altitudes,  the  remainders  are  equal,  that  is,  the 
tetraedrons  A  GHK  and  BLNP  have  equal  altitudes. 


The  tetraedrons  AEIO  and  A  GHK  are  similar.  There- 
fore, EIO,  the  base  of  the  first,  is  to  GHK,  the  base  of  the 
second,  as  the  square  of  the  altitude  of  the  first  is  to  the  square 
of  the  altitude  of  the  second.  For  a  like  reason,  the  base  CDF 
is  to  the  base  LNP  as  the  square  of  the  greater  altitude  is  to 
the  square  of  the  less. 

Therefore,       EIO :  GHK=  CDF'.  iiVP. 

By  alternation, 

EIO :  CDF=  GHK'.LNP. 

Corollaries. — ^I.  "When  the  bases  are  equivalent  the  sections 
are  equivalent. 

II.  When  the  bases  are  equal  the  sections  are  equal.  For 
they  are  similar  and  equivalent. 

28.  Exercises. — 1.  Find  the  area  of  the  surface  of  a  regular  octa- 
gonal pyramid  whose  slant  height  is  5  inches,  and  a  side  of  whose  base  is 
2  inches. 

2.  The  area  of  the  entire  surface  of  a  regular  tetraedron,  the  edge 
being  1  inch,  is  V^  square  inches. 


Art.  28.] 


POLTEBRONS. 


203 


3.  When  two  pyramids  of  equal  altitude  have  their  bases  in  the  same 
plane,  and  are  cut  by  a  plane  parallel  to  their  bases,  the  areas  of  the  sec- 
tions are  proportional  to  the  areas  of  the  bases. 

4.  A  right  prism  has  less  surface  than  any  other  prism  of  equal  base 
and  equal  altitude ;  and  a  regular  prism  has  less  surface  than  any  other 
right  prism  of  equivalent  base  and  equal  altitude,  and  of  same  number  of 
sides. 

5.  A  regular  pyramid  and  a  regular  prism  have  equal  hexagonal  bases, 
and  altitudes  equal  to  three  times  the  radius  of  the  base ;  required  the 
ratio  of  the  areas  of  their  lateral  surfaces. 

6.  In  what  case  of  the  first  proposition  of  Article  22  are  the  figures 
similar,  and  in  what  case  are  they  symmetrically  similar  ? 

7.  Demonstrate  the  Corollary  of  Article  25,  without  using  the 
Theorem  ? 

Measure  of  Volume. 

39.  The  cube,  on  account  of  its  simple  form  and  the 
unity  of  its  three  dimensions,  has  the  same  place  among  solids 
that  the  square  has  among  plane  figures.  It  is  the  unit  of 
volume.  That  is,  whatever  straight  line  is  taken  as  the  unit 
of  length,  the  cube  whose  edge  is  of  that  length  is  the  unit  of 
volume,  as  the  square  whose  side  is  of  that  length  is  the  unit 
of  area. 


Volume  of  Parallelopipeds. 

Theorem. — The  volume  of  a  rectangular  parallelopiped  is 
equal  to  the  product  of  its  length,  breadth,  and  altitude. 

That  is,  the  number  of  cubical  units  contained  in  a  rectangu- 
lar parallelopiped  is  equal  to  the 
product  of  the  numbers  of  linear 
units  in  the  length,  the  breadth, 
and  the  altitude. 

If  the  altitude  AE,  the  length 
EI,  and  the  breadth  10,  have  a 
common  measure,  let  each  be  di- 
vided by  it ;  and  let  planes,  par- 
allel to  the  faces  of  the  prism, 
pass  through  all  the  points  of  division,  B,  (7,  D,  etc. 


\ 
0\ 


^^^^V 


I 


3 


H 


204  GEOMETRY  OF  SPACE.  [Chap.  X. 

All  tlie  angles  formed  by  these  planes  and  their  intersections 
are  right  angles,  and  each  of  the  intercepted  lines  is  equal  to  the 
linear  unit  used  in  dividing  the  edges  of  the  prism.  Therefore, 
the  prism  is  divided  into  equal  cubes.  The  number  of  these  at 
the  base  is  equal  to  the  number  of  rows,  multiplied  by  the  num- 
ber in  each  row ;  that  is,  the  product  of  the  length  by  the 
breadth.  There  are  as  many  layers  of  cubes  as  there  are  linear 
units  of  altitude.  Therefore,  the  whole  number  is  equal  to  the 
product  of  the  length,  breadth,  and  altitude. 

It  follows  that  if  two  rectangular  parallelopipeds  are 
such  that  there  is  a  common  measure  of  all  their  dimensions, 
then  one  volume  is  to  the  other  as  the  product  of  the  dimen- 
sions of  the  first  is  to  the  product  of  the  dimensions  of  the 
second. 

When  the  length,  breadth,  and  altitude  of  the  parallelepiped 
have  no  common  measure,  let  any  line  be  assumed  as  the  unit  of 
length,  and  the  cube  having  this  edge  as  the  unit  of  volume. 
Applying  this  to  the  parallelopiped,  a  part  only  is  measured,  but, 
if  a  cube  whose  edge  is  an  aliquot  part  of  the  unit  of  length  is 
used,  a  greater  part  of  the  parallelopiped  may  be  measured  ;  and, 
by  using  a  unit  regularly  smaller,  the  part  measured  is  a  variable 
parallelopiped  whose  limit  is  the  one  given.  Also  the  product 
of  the  three  dimensions  of  this  variable  is  a  variable  whose  limit 
is  the  product  of  the  dimensions  of  the  given  parallelopiped.  At 
every  step  the  volume  of  the  variable  is  to  the  unit  of  volume  as 
the  product  of  the  dimensions  of  the  variable  to  the  product  of 
the  dimensions  of  the  unit. 

As  these  ratios  are  always  equal,  their  limits  are  equal,  that 
is,  the  volume  of  the  parallelopiped  is  to  the  unit  of  volume  as 
the  product  of  the  three  dimensions  of  the  parallelopiped  is  to 
the  cube  of  unity. 

Therefore  a  rectangular  parallelopiped  is  measured  by  the 
product  of  its  length,  breadth,  and  altitude. 


30,   Theorem. — The  volume  of  any  parallelopiped  is  equal 
to  the  product  of  its  length,  breadth,  and  altitude. 

As  this  has  been  demonstrated  for  the  rectangular  parallel- 


i 


Art.  30.] 


POLYEDRONS. 


205 


opiped,  it  is  sufficient  to  show  that  any  parallelopiped  is 
equivalent  to  a  rectangular  one  having  the  same  linear  dimen- 
sions. 

Let  the  lower  bases  of  the  two  prisms  be  on  the  same  plane. 
Then  their  upper  bases  are  in  one  plane.  Through  every  point 
of  the  altitude  AE  pass  a  plane  parallel  to  the  base  AI,  and  let 
it  cut  both  prisms. 


Now  every  section  in  either  prism  is  equal  to  the  base  ;  but 
the  bases,  having  the  same  dimensions,  are  equivalent.  Therefore 
every  section  in  one  solid  has  its  corresponding  equivalent  section 
in  the  other.  Therefore  the  sum  of  the  sections  in  one  is  equivalent 
to  the  sum  of  the  sections  in  the  other  ;  but  the  volume  of  each 
prism  consists  of  this  sum  of  an  infinite  number  of  plane 
figures. 

Besides  the  above  demonstration  by  the  method  of  in- 
divisibles (IV,  31),  the  theorem  may  be  demonstrated  by 
the  superposition  and  coincidence  of  equal  figures,  as  fol- 
lows : 

Let  AF  be  any  oblique  parallelopiped.  It  is  equivalent  to 
the  parallelopiped  AL,  which 
has  a  rectangular  base,  AH; 
since  the  prism  LHEO  is 
equal  to  the  prism  DGAI. 
But  the  parallelopipeds  AF 
and  AL  have  the  same  length, 
breadth,  and  altitude. 

By  similar  reasoning,  the 
prism  AL  is  shown  to  be 
equivalent  to  a  prism  of  the  same  base  and  altitude,  with  two 


206  GEOMETRY  OF  SPACE.  [Chap.  X. 

of  its  opposite  sides  rectangles.  This  third  prism  is  then  shown 
to  be  equivalent  to  a  fourth,  which  is  quite  rectangular,  and  has 
the  same  dimensions  as  the  others. 

Corollaries. — I.  The  volume  of  any  parallelopiped  is  equal 
to  the  product  of  its  base  by  its  altitude. 

II.  The  volumes  of  any  two  parallelopiped s  are  to  each  other 
as  the  products  of  their  three  dimensions. 

Volume  of  Prisms. 

31.  Theorem. —  The  volume  of  a  triangular  prism  is  equal 
to  the  product  of  its  base  by  its  altitude. 

The  base  of  a  right  triangular  prism  may  be  considered  as 
one  half  of  the  base  of  a  right  parallelopiped.  Then  the  whole 
parallelopiped  is  double  the  given  prism,  for  it  is  composed  of 
two  right  prisms  having  equal  bases  and  the  same  altitude,  of 
which  the  given  prism  is  one  (16,  c).  Therefore,  the  given 
prism  is  measured  by  half  the  product  of  its  altitude  by  the  base 
of  the  parallelopiped ;  that  is,  by  the  product  of  its  own  base 
and  altitude. 

It  remains  to  be  proved  that  an  oblique  triangular  prism  is 
equivalent  to  a  right  triangular  prism  of  the  same  base  and  alti- 
tude. This  may  be  demonstrated  by  the  method  of  indivisibles 
used  in  the  preceding  Article. 

Corollary. — The  volume  of  any  prism  is  equal  to  the  product 
of  its  base  by  its  altitude.  For  any  prism  is  composed  of  trian- 
gular prisms,  having  the  common  altitude  of  the  given  prism, 
the  sum  of  their  bases  forming  the  given  base. 


Volume  of  Tetraedrons. 

33.  Theorem. — Two  tetraedrons  of  equivalent  bases  and  of 
the  same  altitude  are  equivalent. 

Suppose  the  bases  of  the  two  tetraedrons  to  be  in  the  same 
plane.  Through  every  point  of  the  common  altitude  pass  a 
plane  parallel  to  the  base  and  produce  it  through  both 
solids. 

Every  section  in  one  tetraedron  is  equivalent  to  the  section 


Art.  32.]  P0LTEDR0N8,  207 

made  by  the  same  plane  in  the  other  (27,  i).     Therefore  the  sum 
of  all  the  sections  in  one  is  equivalent  to  the  sum  of  all  the  sec- 


tions in  the  other,  but  the  volume  of  each  tetraedron  consists  of 
this  sum  of  an  infinite  number  of  plane  triangles. 

33.  Theorem. — The  volume  of  a  tetraedron  is  equal  to  one 
third  of  the  product  of  the  base  by  the  altitude. 

Upon  the  base  of  any  tetraedron  a  triangular  prism  may  be 
erected,  having  the  same  altitude,  and  one  edge  coincident  with 
an  edge  of  the  tetraedron.  This  prism  may  be  divided  into 
three  tetraedrons,  the  given  one  and  two  others,  which,  taken 
two  and  two,  have  equal  bases  and  equal  altitudes. 

These  three  tetraedrons  are  equivalent ;  therefore  the  volume 
of  the  given  tetraedron  is  one  third  of  the  volume  of  the  prism  ; 
that  is,  one  third  of  the  product  of  its  base  by  its  altitude. 

Volume  of  Pyramids. 

Corollaries. — I.  The  volume  of  any  pyramid  is  equal  to  one 
third  of  the  product  of  its  base  by  its  altitude.  For  any  pyra- 
mid is  composed  of  triangular  pyramids  ;  that  is,  of  tetraedrons 
having  the  common  altitude  of  the  given  pyramid,  and  the  sum 
of  their  bases  forming  the  given  base. 

II.  Symmetrical  prisms  are  equivalent.  The  same  is  true  of 
symmetrical  pyramids. 

III.  The  volume  of  a  frustum  of  a  pyramid  is  found  by  sub- 


208  GEOMETRY  OF  SPACE.  [Chap.  X. 

tracting  the  volume  of  the  pyramid  cut  off  from  the  volume  of 
the  whole. 

When  the  two  bases  and  the  altitude  of  a  frustum  are  given, 
a  rule  for  calculating  the  volume  is  made  as  follows.  Represent 
the  lower  base  by  5^,  the  upper  base  by  a^,  and  the  altitude  by 
A.     Then,  if  x  represents  the  altitude  of  the  part  cut  off. 

Therefore,  x  =  h t and  h-\-x  =  h t .     The    volume 

'  0  —  a  '  o  —  a 

J  a  b^ a^ 

of  the  frustum  is  \b^  h  t \  a^  h  v =:  \  h    ,  = 

J  A  (J2  _|_  j^  _|_  ^2)^  That  is,  the  volume  of  the  frustum  is  equal 
to  the  sum  of  the  volumes  of  three  pyramids  having  the  altitude 
of  the  frustum,  and  bases  respectively  equal  to  its  lower  base, 
its  upper  base,  and  a  mean  proportional  between  them. 

Ratio  of  Similar  Volumes. 

34.  Theorem. — The  volumes  of  similar  polyedrons  are  pro- 
portional to  the  cubes  of  homologous  lines. 

First,  suppose  the  figures  to  be  tetraedrons.  Let  AH  and 
BGhe.  the  altitudes. 

Then  (26), 

EIO  :  CDF=  EI^  :  CF^  =  AH^  :  BG^. 


By  the  proportionality  of  homologous  lines, 

4  AH:  iBG  =  EI'.  CF=  AH.BG, 


Aet.  34.]  P0L7EDR0NS. 

Multiplying  (T,  3,  vii), 

AJEIO  \BCFD  =  EI^  :  CF^  -  AH^  :  BG^\ 

or,  as  the  cubes  of  any  other  homologous  lines. 

Next,  let  any  two  similar  polyedrons  be  divided  into  the  same 
number  of  tetraedrons.  Then,  as  just  proved,  the  volumes  of 
the  homologous  parts  are  proportional  to  the  cubes  of  the  homol- 
ogous lines.  By  arranging  these  in  a  continued  proportion,  as 
in  Article  13,  Chapter  YII,  it  is  shown  that  the  volume  of  one 
polyedron  is  to  the  volume  of  the  other  as  the  cube  of  any  line 
of  the  first  is  to  the  cube  of  the  homologous  line  of  the 
second. 


35.  Exercises. — 1.  If  two  tetraedrons  have  a  triedral  vertex  in 
each  equal,  their  volumes  are  in  the  ratio  of  the  products  of  the  edges 
which  contain  the  equal  vertices. 

2.  The  plane  that  bisects  a  diedral  angle  of  a  tetraedron  divides  the 
opposite  edge  in  the  ratio  of  the  areas  of  the  adjacent  faces. 

3.  The  volume  of  a  prism  is  equal  to  the  product  of  one  of  its  lateral 
edges  by  the  area  of  a  section  perpendicular  to  that  edge. 

4.  What  is  the  ratio  between  the  edges  of  two  cubes,  one  of  which 
has  twice  the  volume  of  the  other  ? 

The  duplication  of  the  cube  was  one  of  the  celebrated  problems  of 
ancient  times.  The  oracle  at  Delphos  demanded  of  the  Athenians  a  new 
altar,  of  the  same  shape,  but  of  twice  the  volume  of  the  old  one.  The 
efforts  of  the  Greek  geometers  were  chiefly  aimed  at  a  graphic  solution ; 
that  is,  the  edge  of  one  cube  being  given,  to  draw  a  line  equal  to  the  edge 
of  the  other.  Whilst  using  no  instruments  but  the  rule  and  compasses, 
they  failed.  The  student  will  find  no  difficulty  in  making  an  arithmeti- 
cal solution,  within  any  desired  degree  of  approximation. 

5.  How  may  a  pyramid  be  cut  by  a  plane  parallel  to  the  base,  so  as  to 
make  the  area  or  the  volume  of  the  part  cut  off  have  a  given  ratio  to  the 
area  or  the  volume  of  the  whole  pyramid? 

6.  The  area  of  the  lower  base  of  a  frustum  of  a  pyramid  is  five  square 
feet,  of  the  upper  base  one  and  four-fifths  square  feet,  and  the  altitude  is 
two  feet;  required  the  volume. 

36.  Scholium. — Among  solid  as  well  as  plane  figures,  simi- 
larity involves  equality  of  homologous  angles  and  equality  of 

10 


''210  GEOMETRY  OF  SPACE.  [Chap.  X. 

the  ratios  of  homologous  lines — ^the  former  is  the  essential  char- 
acter as  expressed  in  the  definition,  and  the  latter  is  a  necessary 
consequence. 

Equality  of  some  corresponding  lines  is  never  a  condition  of 
similarity,  but  it  is  always  a  condition  of  equality,  both  in  plane 
and  in  solid  figures. 

On  the  other  hand,  symmetry  shows  a  great  contrast  be- 
tween figures  on  a  plane  and  figures  in  space.  Plane  symmetri- 
cal figures  are  equal,  because,  by  turning  one,  they  may  coin- 
cide ;  but  symmetrical  solids  may  not  coincide.  Symmetry  in- 
volves equality  of  corresponding  angles,  and  of  corresponding 
straight  lines,  and  of  corresponding  plane  surfaces  ;  also  it  in- 
volves equivalence  both  as  to  area  and  volume. 

When  a  polyedron  is  symmetrical  to  a  second  and  similar  to 
a  third,  the  second  and  third  are  called  symmetrically  similar. 
The  proportion  of  homologous  lines  of  similar  polyedrons  is 
equally  true  of  those  that  are  symmetrically  similar. 

There  are  three  ratios  between  similar  figures  :  viz.,  the 
linear  between  homologous  lengths,  the  superficial  between 
homologous  areas,  and  the  cubic  between  homologous  volumes — 
the  second  and  third  being  respectively  the  second  and  third 
powers  of  the  first. 

The  use  of  the  term  cube  to  designate  the  third  power  arose 
in  this  way. 

In  the  measure  of  an  area  there  are  two  linear  dimensions  ; 
and  in  the  measure  of  a  volume  three  linear,  or  one  linear  and  one 
superficial. 


37.  Miscellaneous  Exercises. — 1.  The  opposite  vertices  of  a 
parallel opiped  are  symmetrical  triedrals. 

2.  The  diagonals  of  a  rectangular  parallelopiped  are  equal. 

3.  The  square  of  the  diagonal  of  a  rectangular  parallelopiped  is  equiv- 
alent to  the  sum  of  the  squares  of  its  length,  breadth,  and  altitude. 

4.  The  diagonals  of  a  parallelopiped  bisect  each  other;  the  lines 
that  join  the  midpoints  of  the  opposite  edges  bisect  each  other;  the  lines 
that  join  the  centers  of  the  opposite  faces  bisect  each  other;  and  the 
point  of  intersection  is  the  same  for  all  these  lines,  and  is  a  center  of 
symmetry. 


Art.  Z1.]  POLYEDROKS.  211 

5.  A  cube  is  the  largest  parallel opiped  of  the  same  extent  of  surface 
(VII,  23). 

6.  The  centroid  of  a  triangle  is  the  point  of  intersection  of  the  rae- 
dials.  The  four  lines  that  join  the  vertices  of  a  tetraedron  to  the  cen- 
troids  of  the  opposite  faces  intersect  each  other  in  one  point.  In  what 
ratio  do  the  lines  just  described  in  the  tetraedron  divide  each  other? 
[Let  a  triangular  diagram  represent  a  plane  section  of  a  tetraedron  made 
through  one  edge  and  the  midpoint  of  the  opposite  edge.] 

7.  Two  different  tetraedrons,  and  only  two,  may  be  formed  with  the 
same  four  triangular  faces ;  and  these  two  tetraedrons  are  symmetrical. 
[Use  paper  models.] 

8.  In  any  polyedron  the  sum  of  the  number  of  vertices  and  the  num- 
ber of  faces  exceeds  by  two  the  number  of  edges. 

9.  In  what  way  is  each  of  the  five  regular  polyedrons  symmetrical  ? 


CHAPTER  XL 


SOLIDS  OF  REVOLUTION, 


Article  1. — Solids  of  Revolution  are  generated  by  the 
revolution  of  a  plane  figure  about  an  axis. 

A  Cone  is  a  solid  described  by  the  revolution  of  a  right- 
angled  triangle  about  one  of  its  sides  as 
an  axis.  The  other  side  describes  a  plane 
surface,  a  circle,  having  for  its  radius  the 
line  by  which  it  is  described.  The  hy- 
potenuse describes  a  curved  surface. 

The  plane  surface  of  a  cone  is  called 
its  base.  The  opposite  extremity  of  the 
axis  is  the  vertex.  The  altitude  is 
the  distance  from  the  vertex  to  the  base, 
and  the  slant  height  is  the  distance 
from  the  vertex  to  the  circumference  of  the  base. 

A  Cylinder  is  a  solid  described  by 
the  revolution  of  a  rectangle  about  one 
of  its  sides  as  an  axis.  The  sides  per- 
pendicular to  the  axis  describe  circles ; 
the  opposite  side  describes  a  curved  sur- 
face. 

The  plane  surfaces  of  a  cylinder  are 
called  its  bases,  and  the  perpendicular 
distance  between  them  is  its  altitude. 

These  figures  are  a  cone  and  a  cylin- 
der of  revolution  ;  other  cones  and  cylin- 
ders are  not  usually  discussed  in  Elementary  Geometry. 


Art.  1.]  SOLIDS  OF  REVOLUTION.  213 

A  Sphere  is  a  solid  described  by  the  revolution  of  a  semi- 
circle about  its  diameter  as  an  axis. 

The  center,  radius,  and  diame- 
ter of  the  sphere  are  the  same  as 
those  of  the  generating  circle.  The 
spherical  surface  is  described  by  the 
circumference. 

Corollaries, — I.  Radii  of  the  same 
sphere  are  equal.  The  same  is  true  of 
the  diameters. 

II.  Spheres  having  equal  radii  are 
equal. 

III.  A  plane  passing  through  the  center  of  a  sphere  divides 
it  into  equal  parts.  The  halves  of  a  sphere  are  called  hemi- 
spheres. 

IV.  A  conical  surface  is  the  locus  of  those  points  in  space 
the  distance  of  each  of  which  from  a  straight  line  given  in  posi- 
tion (the  axis)  has  a  constant  ratio  to  the  distance  from  a  given 
point  of  that  line  (the  vertex).  This  constant  ratio  is  that  of  the 
radius  of  the  base  to  the  slant  height  of  the  cone. 

y.  A  cylindrical  surface  is  the  locus  of  those  points  that 
are  at  a  uniform  distance  from  a  straight  line  given  in 
position  (the  axis).  This  distance  is  equal  to  the  radius  of 
the  base. 

VI.  A  spherical  surface  is  the  locus  of  those  points  that  are 
at  a  uniform  distance  from  a  given  point  (the  center).  This  dis- 
tance is  the  radius  of  the  sphere. 

As  the  curved  surfaces  of  the  cone  and  cylinder  are  gener- 
ated by  the  motion  of  a  straight  line,  both  of  these  surfaces  are 
straight  in  one  direction. 

A  straight  line  from  the  vertex  of  a  cone  to  the  circumfer- 
ence of  the  base  lies  wholly  in  the  surface.  So  a  straight  line 
perpendicular  to  the  base  of  a  cylinder  at  its  circumference  lies 
wholly  in  the  surface.  For,  in  each  case,  the  position  had  been 
occupied  by  the  generating  line. 

One  surface  is  tangent  to  another  when  it  meets,  but  being 
produced  does  not  cut  it.  The  place  of  contact  of  a  plane  with 
a  conical  or  cylindrical  surface  is  a  straight  line  ;  since,  from 


214 


GEOMETRY  OF  SPACE. 


[Chap.  XI. 


any  point  of  one  of  those  surfaces,  it  is  straight  in  one  direc- 
tion. 

3,  Theorem.— r A  plane  that  is  perpendicular  to  a  radius  of 
a  sphere  at  its  extremity  is  tangent  to  the  sphere. 

If  straight  lines  extend  from 
the  center  of  the  sphere  to  any- 
other  point  of  the  plane,  they  are 
oblique  and  longer  than  the  radius, 
which  is  perpendicular  (IX,  8,  i). 
Therefore,  every  point  of  the  plane 
except  one  is  beyond  the  surface 
of  the  sphere,  and  the  plane  is  tan- 
gent. 

Corollary. — A  spherical  sur- 
face is  curved  in  every  direction. 
Every  possible  section  of  it  is  a  curve. 

3.  Scholium. — Models  of  cones,  of  frusta,  and  of  cylinders 
may  be  made  from  paper,  taking  a  sector  of  a  circle  for  the 
curved  surface  of  a  cone,  and  a  rectangle  for  the  curved  surface 
of  a  cylinder. 

The  curvature  of  the  sphere  in  every  direction  renders  it  im- 
possible to  construct  an  exact  model  with  plane  paper.  But  the 
student  is  advised  to  procure  or  make  a  globe,  upon  which  he 
can  draw  diagrams,  representing  spherical  figures  better  than  it 
is  possible  to  do  on  a  plane  surface. 


Secant  Planes. 

4.  Every  point  of  the  line  that  describes  the  curved  surface 
of  a  cone,  or  of  a  cylinder,  moves  in  a  plane  parallel  to  the  base 
(IX,  20).  Therefore,  if  a  cone  or  a  cylinder  is  cut  by  a  plane 
parallel  to  the  base,  the  section  is  a  circle. 

A  Frustum  of  a  cone  is  that  part  of  the  cone  between  the 
base  and  a  plane  parallel  to  the  base. 

"When  a  conical  surface  is  cut  by  a  plane,  the  form  of  the 


Art.  4.]  SOLIDS  OF  REVOLUTION.  215 

line  of  intersection  varies  according  to  the  relative  position  of 
the  plane  and  the  axis  of  the  cone.  It  may  be  a  circumference 
of  a  circle,  an  ellipse,  a  parabola,  a  hyperbola,  or  it  may  be  two 
straight  lines. 

These  Conic  Sections  are  not  considered  in  Elementary 
Geometry.  Their  properties  are  usually  investigated  by  the 
application  of  algebra. 

If  a  c}^inder  is  cut  by  a  plane  obliquely,  the  intersection  of 
the  plane  and  the  curved  surface  is  an  ellipse. 

Since  every  point  in  the  arc  that  describes  a  spherical  sur- 
face moves  in  a  plane  perpendicular  to  the  axis,  every  plane  sec- 
tion perpendicular  to  the  axis  is  a  circle.  Since  every  point  in 
the  spherical  surface  is  at  the  same  distance  from  the  center,  any 
part  of  the  surface  may  coincide  with  any  other  part  so  far  as 
they  both  extend,  and  any  diameter  may  be  considered  as  the 
axis  by  which  the  figure  was  described. 

Corollaries. — I.  Every  plane  section  of  a  sphere  is  a  circle, 
for  every  plane  is  perpendicular  to  some  axis. 

11.  Three  points  on  a  spherical  surface  determine  a  circum- 
ference which  lies  in  that  surface. 


Similar  Solids  of  Revolution, 

5.  Theorem. — Solids  of  revolution,  described  hy  the  revolu- 
tion of  similar  plane  figures  about  homologous  lines  are  simi- 
lar. 

The  generating  figures  being  so  placed  that  they  have  a 
common  axis,  they  have  a  center  of  similarity  on  this  axis. 

Then,  whatever  line  is  made  in  one,  there 
is  a  parallel  homologous  line  in  the  other, 
making  equal  angles. 

Corollaries. — I.  Cones  are  similar  when 
the  ratio  of  the  radius  of  the  base  to  the 
slant  height  is  the  same. 

II.  If  a  cone  is  cut  by  a  plane  parallel  to 
the  base,  the  cone  cut  off  is  similar  to  the 
whole. 


216 


GEOMETRY  OF  SPACE. 


[Chap.  XI. 


III.  Cylinders  are  similar  when  the  ratio 
of  the  diameter  to  the  altitude  is  the 
same. 

IV.  All  spheres  are  similar,  for  all  semi- 
circles are  similar. 


V.  In  whatever  position  they  are  placed,  two  spheres  have 
an  internal  and  an  external  center  of  similarity. 


*i^ 


Cones. 

6.  A  cone  is  said  to  be  inscribed  in  a  pyramid  when  their 
bases  lie  in  one  plane,  and  the  sides  of  the  pyramid  are  tangent 
to  the  curved  surface  of  the  cone.  The  pyramid  is  said  to  be 
circumscribed  about  the  cone. 

A  cone  is  said  to  be  circumscribed  about  a  pyramid  when 
their  bases  lie  in  one  plane,  and  the  lateral  edges  of  the  pyramid 
lie  in  the  curved  surface  of  the  cone.  Then  the  pyramid  is  in- 
scribed in  the  cone. 

Theorem. — A  cone  is  the  limit  of  the  variable  pyramid  that 
is  produced  when  the  number  of  sides  of  an  inscribed  or  of  a  cir- 
cumscribed pyramid  is  doubled  continually,  bisecting  the  arcs  at 
the  base  at  every  step  of  the  variation. 

The  base  of  the  cone  is  a  circle,  which  is  the  limit  of  the 
variable  polygons  that  are  the  bases  of  the  inscribed  and  cir- 
cumscribed pyramids.  At  every  step  of  the  variation,  the  sides 
of  the  circumscribed  polygon  are  tangent  to  the  circle  ;  hence 
every  side  of  the  pyramid  is  tangent  to  the  curved  surface,  and 
the  variable  pyramid  is  always  circumscribed.     The  vertices  of 


Art.  6.]  SOLIDS  OF  REVOLUTION.  217 

the  polygon  inscribed  in  the  base  are  always  in  the  circumfer- 
ence, the  lateral  edges  of  this  variable  pyramid  must  at  every 


step  lie  in  the  curved  surface,  and  this  variable  pyramid  is  al- 
ways inscribed  in  the  cone. 

It  has  been  demonstrated  (YIII,  2)  that  the  circumscribed 
and  inscribed  polygons  can  approach  each  other  within  less  than 
any  assignable  difference  of  surface.  It  follows  that  the  varia- 
ble circumscribed  and  inscribed  pyramids  can  approach  each 
other  indefinitely.  As  the  cone  is  always  between  them,  it  must 
be  the  limit  of  both. 

Corollaries. — I.  The  area  of  the  curved  surface  of  a  cone  is 
equal  to  half  the  product  of  the  slant  height  by  the  circumfer- 
ence of  the  base.  Also,  it  is  equal  to  the  product  of  the  slant 
height  by  the  circumference  of  a  section  midway  between  the 
vertex  and  the  base.     It  may  be  expressed  by  'nRH. 

II.  The  volume  of  a  cone  is  equal  to  one  third  of  the  prod- 
uct  of  the   base  by  the   altitude.      It   may  be   expressed  by 

III.  The  area  of  the  curved  surface  of  the  frustum  of  a  cone 
is  equal  to  half  the  product  of  its  slant  height  by  the  sum  of  the 
circumferences  of  its  bases.  Also,  it  is  equal  to  the  product  of 
its  slant  height  by  the  circumference  of  a  section  midway  be- 
tween the  bases. 


218 


GEOMETRY  OF  SPACE. 


[Chap.  XL 


Cylinders. 

7.  A  cylinder  is  inscribed  in  a  prism  or  circumscribed  about 
it  in  the  same  way  that  a  cone  is  inscribed  or  circumscribed 
about  a  pyramid. 


;7^ 


Theorem. — A  cylinder  is  the  limit  of  the  variable  prism  that 
is  produced  when  the  number  of  sides  of  an  inscribed  or  circum- 
scribed prism  is  doubled  continually,  bisecting  the  arcs  at  the 
base  at  each  step  of  the  variation. 

The  demonstration  is  similar  to  the  last. 

Corollaries, — I.  The  area  of  the  curved  surface  of  a  cylinder 
is  equal  to  the  product  of  the  altitude  by  the  circumference  of 
the  base.     It  may  be  expressed  by  2r:ItA. 

II.  The  volume  of  a  cylinder  is  equal  to  the  product  of  the 
base  by  the  altitude.     It  may  be  expressed  by  ttR^A. 


Spheres. 

8.  A  sphere  is  said  to  be  inscribed  in  a  polyedron  when 
the  faces  are  tangent  to  the  curved  surface  ;  and  the  polyedron 
is  circumscribed  about  the  sphere.  A  sphere  is  circum- 
scribed about  a  polyedron  when  the  vertices  all  lie  in  the 
curved  surface  ;  and  the  polyedron  is  inscribed  in  the  sphere. 

Thus  the  center  of  the  sphere  is  equally  distant  from  all  the 
faces  of  the  circumscribed  polyedron.  It  is  also  equally  distant 
from  all  the  vertices  of  the  inscribed  polyedron. 

Let  ASF  be  a  right-angled  triangle  and  BFD  a  semicircle, 
the  hypotenuse  AF  being  a  secant,  and  the  vertex  F  in  the  cir- 
cumference. From  -EJ  the  point  where  AF  cuts  the  arc,  let  a 
perpendicular  EG  fall  upon  AD. 


Art.  8.] 


SOLIDS  OF  REVOLUTION. 


219 


Suppose    this    figure    to   revolve   about   AD.      The   trian- 
gle AFH  describes  a  cone,  the  trapezoid 
EGHF  describes  the  frustum  of  a  cone, 
and  the  semicircle  describes  a  sphere. 

The  points  E  and  F  describe  the  cir- 
cumferences of  the  bases  of  the  frustum  ; 
and  these  circumferences  lie  in  the  surface 
of  the  sphere. 

A  frustum  of  a  cone  is  said  to  be  in- 
scribed in  a  sphere  when  the  circumfer- 
ences of  its  bases  lie  in  the  surface  of  the 
sphere. 

A  Great  Circle  of  a  sphere  is  a  section  made  by  a  plane 
through  the  center. 

A  Small  Circle  of  a  sphere  is  a  section  made  by  a  plane 
not  through  the  center. 

Corollaries. — The  following  are  derived  from  the  definitions 
and  properties  of  the  sphere,  regular  polyedrons  (X,  11,  c),  and 
other  figures. 

I.  All  great  circles  of  a  sphere  are  equal,  having  the  same 
radius  as  the  sphere. 

II.  A  great  circle  of  a  sphere  is  larger 
than  a  small  circle.  For  a  plane  through 
the  diameter  of  a  small  circle  and  the  cen- 
ter of  the  sphere  is  the  plane  of  a  great 
circle,  which  has  the  diameter  of  the  small 
circle  for  a  chord. 

III.  Of  two  unequal  small  circles  of  a 
sphere,  the  greater  one  is  nearer  the  cen- 
ter.    Equal  small  circles  are  equally  distant  from  the  center. 

lY.  Two  great  circles  bisect  each  other,  and  their  intersec- 
tion is  a  diameter  of  the  sphere. 

V.  The  axis  of  every  circle  passes  through  the  center  of  the 
sphere. 

The  Poles  of  a  circle  are  the  points  where  its  axis  pierces 
the  spherical  surface. 

VI.  Circles  whose  planes  are  parallel  to  each  other  have  the 
same  axis  and  the  same  poles. 


220  GEOMETRY  OF  SPACE,  [Chap.  XI. 

VIL  Through  any  three  points  on  the  surface  a  circumfer- 
ence of  a  small  circle  may  pass,  provided  the  center  of  the  sphere 
is  not  in  the  plane  of  the  points  given. 

VIII.  If  two  points  on  the  surface  are  diametrically  opposite, 
any  number  of  arcs  of  great  circles  may  join  them,  all  of  which 
are  semi-circumferences.  In  any  other  position  two  points  on 
the  surface  determine  one  great  circle. 

IX.  A  sphere  may  be  circumscribed  about  any  polyedron 
that  has  a  point  equally  distant  from  all  its  vertices. 

X.  A  spherical  surface  may  pass  through  any  four  points 
that  are  not  all  in  one  plane. 

XI.  A  sphere  may  be  inscribed  in  any  polyedron  that  has  a 
point  equally  distant  from  all  its  faces. 

XII.  Every  face  of  an  inscribed  polyedron  may  have  a  circle 
circumscribed  about  it ;  for  the  section  of  the  sphere  made  by 
the  plane  of  the  face  is  such  a  circle. 

XIII.  The  points  of  contact  of  two  adjacent  faces  of  a  cir- 
cumscribed polyedron  are  equally  distant  from  the  edge  that  is 
between  those  faces. 

XI Y.  The  points  of  contact  of  the  faces  that  constitute  one 
polyedral  vertex  of  a  circumscribed  polyedron  are  equally  dis- 
tant from  that  vertex  ;  for  the  point  of  coutact,  the  polyedral 
vertex,  and  the  center  of  the  sphere,  are  the  vertices  of  a  right- 
angled  triangle,  and  all  such  triangles  for  one  polyedral  vertex 
are  equal. 

XV.  The  points  of  contact  of  all  those  faces  of  a  circum- 
scribed polyedron  that  constitute  one  polyedral  vertex  lie  in  one 
plane. 

XVI.  The  points  just  mentioned  may  be  the  vertices  of  one 
face  of  an  inscribed  polyedron. 

XVII.  The  remaining  faces  of  an  inscribed  polyedron  being 
made  in  the  same  way,  in  two  such  polyedrons,  one  circum- 
scribed and  one  inscribed  in  a  sphere,  every  vertex  of  one  cor- 
responds to  a  face  of  the  other. 

XVIII.  The  centers  of  the  faces  of  a  cube  being  the  vertices 
of  a  regular  octaedron,  a  sphere  that  is  inscribed  in  the  cube  is 
circumscribed  about  the  octaedron. 

XIX.  The  centers  of  the  faces  of  a  regular  octaedron  being 


Art.  8]  SOLIDS  OF  REVOLUTION,  221 

the  vertices  of  a  cube,  a  sphere  that  is  inscribed  in  the  octaedron 
is  circumscribed  about  the  cube. 


9.  Exercises. — 1.  A  section  of  a  cone,  by  a  plane  passing  through 
the  vertex,  is  an  isosceles  triangle. 

2.  A  section  of  a  cylinder  made  by  a  plane  perpendicular  to  the  base 
is  a  rectangle. 

3.  If  a  plane  is  tangent  to  a  sphere  at  a  point  on  the  circumference  of 
a  section  made  by  a  second  plane,  the  intersection  of  these  planes  is  tan- 
gent to  that  circumference. 

4.  Of  all  the  small  circles  through  two  given  points  on  the  surface  of 
a  sphere,  the  smallest  is  that  which  has  for  its  diameter  the  straight  line 
joining  the  given  points. 

5.  Can  every  regular  pyramid  have  a  sphere  circumscribed?  In- 
scribed ? 

6.  Can  a  sphere  be  circumscribed  about  every  regular  prism?  In- 
scribed ? 

7.  Show  upon  what  definitions  or  principles  the  corollaries  in  the  last 
Article  depend. 

Spherical  Arcs  and  Angles. 

10.  There  are  some  remarkable  analogies  and  some  striking 
contrasts  between  lines  made  on  the  surface  of  a  sphere  and 
those  made  on  a  plane.  Distance  is  usually  measured  along  a 
straight  line,  but  any  path  on  a  spherical  surface  must  be  a 
curve  (2,  c). 

Theorem, —  Of  all  the  arcs  of  circles  from  one  point  to 
another  on  the  surface  of  a  sphere,  the  shortest  is  the  minor  arc 
of  the  great  circle,  and  the  longest  is  the  major  arc  of  the  great 
circle. 

For,  when  several  arcs  have  a  common  chord,  the  minor  arc 
of  the  circle  having  the  greatest  radius  is  the  shortest,  and  the 
major  arc  of  the  same  circle  is  the  longest  (VIII,  6,  iv). 

CoroUsbries, — I.  Distance  on  a  spherical  surface  is  estimated 
along  the  minor  arc  of  a  great  circle.  If  two  points  are  diamet- 
rically opposite,  none  but  the  arc  of  a  great  circle  can  join  them, 
and  the  distance  is  a  semi-circumference  (8,  viii). 

11.  The  pole  of  a  circle  is  at  the  same  distance  from  every 


222  GEOMETRY  OF  SPACE.  [Chap.  XI. 

point  of  the  circumference,  whether  the  distances  are  measured 
by  chords  or  on  the  surface. 

The  Polar  Radius  of  a  spherical  circle  is  the  distance  along 
the  surface  from  the  pole  to  the  curve. 

III.  If  two  circles  are  equal,  their  polar  radii  are  equal. 

IV.  The  polar  radius  of  a  great  circle  is  a  quadrant. 

V.  If  a  point  on  the  surface  is  at  the  distance  of  a  quadrant 
from  two  points  of  an  arc  of  a  great  circle  which  are  not  diamet- 
rically opposite  to  each  other,  that  point  is  a  pole  of  the  circle 
(IX,  5). 

VI.  Of  two  points  on  the  surface,  one  of  which  is  on  the  cir- 
cumference of  a  great  circle,  the  one  not  on  the  circumference 
is  nearer  to  the  axis  of  the  circle. 

In  the  following  Articles  distance  is  reckoned  on  the  surface, 
unless  otherwise  stated. 


11.  Application. — The  equator  is  a  great  circle  of  the  earth, 
which  is  nearly  a  sphere.  The  parallels  of  latitude  are  small  circles, 
having  the  same  poles  as  the  equator.  The  meridians  are  great  circles 
perpendicular  to  the  equator. 

The  theorem  of  Article  10  is  applied  in  navigation.  A  vessel  crossing 
the  ocean  by  the  shortest  route  from  one  port  to  another,  in  the  same 
latitude,  does  not  sail  along  the  parallel  of  latitude,  for  that  is  an  arc  of  a 
small  circle. 


13.  Two  straight  lines,  having  a  common  point,  can  not 
meet  again,  to  whatever  distance  they  are  produced  ;  and,  the 
longer  they  are,  the  farther  apart  are  the  ends.  On  a  spherical 
surface,  two  lines  as  nearly  straight  as  possible,  that  is,  two  arcs 
of  great  circles,  proceeding  from  one  point,  meet  again  if  pro- 
duced to  the  opposite  point  of  the  sphere,  and  return  eventually 
to  the  point  of  starting. 

A  Spherical  Angle  is  the  difference  in  the  directions  of 
two  arcs  of  great  circles  at  their  point  of  meeting. 

Since  the  direction  of  an  arc  at  a  given  point  is  the  direction 
of  a  straight  line  tangent  to  the  arc  at  that  point,  a  spherical 
angle  is  the  same  as  the  angle  formed  by  lines  tangent  to  the 
given  arcs  at  their  point  of  meeting.     Thus,  the  spherical  angle 


Art.  12.] 


SOLIDS  OF  REVOLUTION. 


223 


DB  G  is  the  same  as  the  angle  HBK^  the  lines  HB  and  BK 
being  respectively  tangent  to  the 
arcs  BD  and  B  G. 

Corollaries. — I.  A  spherical 
angle  is  the  same  as  the  diedral 
angle  formed  by  the  planes  of  the 
two  arcs.  For  the  intersection 
BF  of  the  planes  of  the  arcs  is  a 
diameter,  the  tangents  IIB  and 
KB  are  perpendicular  to  it,  and 
their  angle  is  the  same  as  the  die- 
dral. 

II.  A  spherical  angle  is  equal  to  the  angle  made  by  the  same 
arcs  on  the  opposite  side  of  the  sphere.  Thus,  the  angles  B  and 
F  are  equal. 

III.  A  spherical  angle  is  equal  in  angular  quantity  to  the  arc 
of  a  circle  included  between  the  arms  of  the  angle,  the  pole  of 
the  arc  being  at  the  vertex.  For  such  an  arc,  as  J)  6r,  has  its 
center  on  the  line  BF,  and  its  curvature  is  equal  to  the  diedral 
BF 

lY.  If  two  arcs  of  great  circles  are  perpendicular  to  each 
other,  each  passes  through  the  poles  of  the  other  (IX,  14,  i). 

V.  If  two  arcs  of  great  circles  are  perpendicular  to  a  third, 
they  meet  at  the  poles  of  the  third. 


13.  Theorem. — From  a  point  on  the  surface  of  a  sphere 
there  are  two  perpendiculars  to 
the  circumference  of  a  great  cir- 
cle ;  one  of  which  is  the  shortest, 
and  the  other  the  longest,  arc  that 
can  extend  from  .the  given  point 
to  the  given  circumference. 

Let  BFJD  be  the  great  circle, 
0  the  center  of  the  sphere,  and 
A  the  point  on  the  surface.  Make 
the  plane  BACO  perpendicular 
to  the  circle  BFJD. 

Then  the  arcs  A  C  and  AB 


224 


GEOMETRY  OF  SPACE. 


[Chap.  XL 


are  arcs  of  a  great  circle  and  both  are  perpendicular  to  the  curve 
JBFD. 

Make  AH  perpendicular  to 
the  plane  DCB,  Then  H  is  on 
the  diameter  JBG\  the  nearest 
point  to  H  on  the  given  circum- 
ference is  C,  and  the  farthest 
from  JS  is  B,  Therefore  the 
chord  AG  i^  shorter  and  AB  \^ 
longer  than  a  chord  from  A  to 
any  other  point  of  the  curve 
BFD  (IX,  8,  ii).  Therefore,  of 
all  the  arcs  of  great  circles  from 
A  to  points  of  the  given  circum- 
ference, the  shortest  \^  AC  and  the  longest  is  AB. 

Corollaries. — I.  From  the  pole  of  a  great  circle  any  number 
of  perpendiculars  extend  to  the  circumference. 

II.  The  two  perpendiculars  from  one  point  to  a  circumfer- 
ence of  a  great  circle  together  make  a  semi-circumference  through 
the  pole. 

III.  Distance  between  a  point  on  the  surface  and  an  arc  is 
reckoned  by  the  shorter  perpendicular. 

IV.  Two  points  on  the  surface  at  the  same  distance  from  the 
plane  of  a  circle  are  also  equally  distant  from  the  circumference. 
For  the  arcs  on  the  surface  subtend  equal  angles  at  the  center  of 
the  sphere  (V,  14,  c). 

V.  If  a  point  on  the  surface  is  at  the  same  distance  from  the 
planes  of  two  great  circles,  it  is 

also  equidistant  from  the  circum- 
ferences. 

YI.  If  two  points  on  the  sur- 
face are  at  unequal  distance  from 
the  plane  of  a  circle,  the  point 
farther  from  the  plane  is  also  far- 
ther from  the  circumference. 

VII.  Every  point  of  the  arc 
bisecting  a  spherical  angle  is  equi- 
distant from  the  arms  of  the  an- 


Art.  13.]  SOLIDS  OF  REVOLUTION,  225 

gle.     A  plane  bisecting  the  diedral  -Si^cuts  the  spherical  surface 
in  an  arc  that  bisects  the  angle  B. 

VIII.  The  arms  of  a  spherical  angle  are  farthest  apart  at  a 
quadrant's  distance  from  the  vertex  ;  for  that  point  of  the  arc  is 
farthest  from  the  edge  of  the  diedral.  An  arc  of  a  great  circle 
that  is  perpendicular  to  the  diameter  BF  bisects  the  arcs  B  GF 
and  BDF  at  right  angles,  and  its  length  is  the  maximum  dis- 
tance between  the  arms  of  the  angle  B, 


Spherical  Surfaces. 

14.  A  Luue  is  the  part  of  the  surface  of  a  sphere  between 
two  halves  of  great  circles. 

That  part  of  the  sphere  between  the  two  planes  is  called  a 
spherical  wedge.  Hence,  two  great  circles  divide  the  surface 
into  four  lunes,  and  the  sphere  into  four  wedges. 

A  Zone  is  a  part  of  the  surface  of  a  sphere  between  two 
parallel  planes.  That  part  of  the  sphere  itself  is  called  a  seg- 
ment. The  circular  sections  are  the  bases  of  the  segment,  and 
the  distance  between  the  parallel  planes  is  the  altitude  of  the 
zone  or  segment. 

One  of  the  parallel  planes  may  be  a  tangent,  in  which  case 
the  segment  has  one  base. 

A  spherical  Sector  is  the  part  of  a  sphere  described  by  the 
revolution  of  a  circular  sector  about  a 
diameter  of  the  circle.     It  may  have  ^^Jl 

two  or  three  curved  surfaces. 

If  AB  is  the  axis,  and  the  generat- 
ing sector  is  AFC,  the  sector  has  one 
spherical  and  one  conical  surface  ;  but, 
if,  with  the  same  axis,  the  generating 
sector  is  FCG,  the  sector  has  one  spher- 
ical and  two  conical  surfaces. 

A  Spherical  Polygon  is  part  of 
the  surface  of  a  sphere  included  between  three  or  more  arcs  of 
great  circles. 

Let  (7,  the  center  of  a  sphere,  be  also  the  vertex  of  a  polye- 


226  GEOMETRY  OF  SPACE.  [Chap.  XI. 

dral.     Then  the  planes  of  the  faces  cut  the  surface  of  the  sphere 
in  arcs  of   great  circles,  which  form  a 

polygon  as  BDFGH.     Conversely,  if  a  ^ -^ 

spherical  polygon  has  the  planes  of  its  /  -^    \ 

several  sides  produced,  they  form  a  poly-  /  f\  /\f  \ 
edral  whose  vertex  is  at  the  center  of  I  l.-""''P/  j 
the  sphere.  \    ^^y       J 

Each   angle   of   the  polygon   is  the  ^'^^^^.^-^ 

same  as  the  corresponding  diedral  of  the 
polyedral,  and  each  side  of  the  polygon 

is  the  same  in  curvature  or  angular  quantity  as  the  correspond- 
ing face  of  the  polyedral. 

CoroUa,ries, — I.  The  limit  of  the  sum  of  the  sides  of  a  con- 
vex polygon  is  the  same  as  of  the  faces  of  a  convex  polyedral 
(IX,  41)  ;  and  the  limit  of  the  sum  of  the  angles  is  the  same  as 
of  the  diedrals  of  a  polyedral  (IX,  42). 

The  Spherical  Excess  of  a  polygon  is  the  excess  of  the 
sum  of  its  angles  over  the  sum  of  the  angles  of  a  plane  polygon 
of  the  same  number  of  sides.  If  the  polygon  is  concave,  the  re- 
flex angle  is  counted,  as  in  plane  figures. 

II.  The  spherical  excess  of  a  convex  polygon  is  less  than  four 
right  angles. 

A  Spherical  Pyramid  is  part  of  a  sphere  included  between 
a  spherical  polygon  and  its  corresponding  polyedral.  The  poly- 
edral is  its  base. 

Spherical  Triangles. 

15.  Three  great  circles  divide  the  surface  of  a  sphere  into 
eight  triangles,  as  three  planes  divide  the  space  about  a  point 
into  eight  parts  (IX,  29).  Thus  every  spherical  triangle  has 
seven  others,  whose  elements  (sides  and  angles)  are  either  equal 
or  supplementary  to  those  of  the  given  triangle. 

The  one  opposite  the  given  triangle,  as  GKH  to  FDB^  has 
elements  respectively  equal  to  the  elements  of  the  given  trian- 
gle, but  arranged  in  reverse  order,  the  corresponding  triedrals 
being  symmetrical. 

As  defined  by  some  geometers,  the  sides  of  a  spherical  trian- 


Art.  15.] 


SOLIDS  OF  REVOLUTION-. 


227 


gle  may  exceed  a  semi-circumference.  For  example,  the  figure 
bounded  by  the  arcs  JBKHD, 
DF,  and  FB  is  called  a  spheri- 
cal triangle.  The  properties  of 
such  figures  are  readily  deduced 
from  those  of  triangles  whose 
perimeters  are  less  than  180°. 

The  sides  and  angles  of  a 
spherical  triangle  are  equal  in 
angular  quantity  to  the  faces 
and  diedrals  of  the  correspond- 
ing triedral  at  the  center  of  the 
sphere.     Hence  a  spherical  tri-  ^ 

angle  may  be  isosceles,  or  it  may 

be  equilateral,  or  rectangular,  etc.  Also  two  triangles  may  be 
symmetrical,  or  they  may  be  supplementary. 

The  planes  of  three  great  circles,  each  of  which  is  perpendic- 
ular to  the  other  two,  make  eight  equal  trirectangular  triedrals 
at  the  center  of  the  sphere.  The  eight  corresponding  triangles 
have  their  angles  all  right  angles  and  their  sides  all  quad- 
rants. 

A  Quadrant al  triangle  is  a 
spherical  triangle  whose  sides 
are  all  quadrants.  It  is  also  tri- 
rectangular. Its  area  is  one 
eighth  that  of  the  sphere.  Its 
spherical  excess  is  one  right  an- 
gle. 

Spherical  triangles  differ 
from  triedrals  in  this  respect — 
a  triangle  has  a  certain  area, 
and  its  perimeter  has  a  certain 

length,  a  triangle  is  a  magnitude.  The  elements  of  a  triedral 
are  angular  quantities.  Two  triangles  can  not  be  equal,  unless 
upon  the  same  sphere  or  upon  equal  spheres. 

If  two  triangles  on  unequal  spheres  have  their  elements  re- 
spectively equal  in  angular  quantity,  and  arranged  in  the  same 


228 


GEOMETRY  OF  SPACE, 


[Chap.  XI. 


order,  the  triangles  are  similar.     Placing  the  spheres  concentric- 
ally, the  demonstration  is  evident. 

Corollaries, — I.  The  sum  of  the  angles  of  a  spherical  trian- 
gle is  greater  than  two  and  less  than  six  right  angles  (IX, 
38). 

II.  An  isosceles  spherical  triangle  is  equal  to  its  symmetrical. 
It  has  equal  angles  opposite  the  equal  sides,  and  conversely  (IX, 
38,  c). 

The  radius  being  the  same,  when  two  triangles  have  the  fol- 
lowing elements  respectively  equal,  the  remaining  elements  are 
respectively  equal : 

III.  The  three  sides  (IX,  35),  or 

IV.  The  three  angles  (IX,  86),  or 

V.  Two  sides  and  the  included  angle  (IX,  87),  or 

VI.  One  side  and  the  adjacent  angles  (IX,  38). 

VII.  In  every  case  of  equal  elements,  when  the  arrangement 
is  the  same  the  triangles  are  equal ;  when  reversed,  they  are 
symmetrical. 

VIII.  If  one  arc  of  a  great  circle  bisects  another  perpendicu- 
larly, every  point  of  the  first  is  equidistant  from  the  ends  of  the 
second  arc. 

IX.  From  one  point  on  the 
surface  four  oblique  arcs  make 
equal  angles  with  the  same  cir- 
cumference. If  the  arc  AG 
bisects  DF  at  right  angles, 
the  angles  ADC,  AFC,  AFC, 
and  AGO  are  equal. 

X.  Four  unequal  right-an- 
gled triangles  may  have  three 
elements  respectively  equal. 
For  instance,  the  triangles 
ADG,AGF,AGF,aTidAGG. 

There  are  several  similar  cases  of  unequal  triangles  that  have 
three  elements  respectively  equal ;  some  of  which  are  the  same 
as  analogous  cases  in  plane  triangles. 

XL  Bisectors  of  the  angles  of  a  triangle  meet  at  a  common 
point,  which  is  equally  distant  from  the  sides.     If  perpendicu- 


Art.  15.]  SOLIDS  OF  REVOLUTION.  229 

lars  are  made  from  this  point  to  the  several  sides  and  a  small 
circle  is  made  through  the  feet  of  these  perpendiculars  (8,  vii), 
this  circle  is  inscribed  in  the  triangle.  Compare  Article  3,  Chap- 
ter V. 

XII.  The  three  arcs  that  bisect  perpendicularly  the  sides  of 
a  triangle  meet  in  one  point,  which  is  equally  distant  from  the 
vertices,  and  is  the  pole  of  a  small  circle  circumscribing  the  tri- 
angle. 

XIII.  If  two  triangles  are  equal  or  symmetrical,  the  circum- 
scribed circles  are  equal. 

16.  Theorem. — In  a  spherical  triangle^  the  greater  side  is 
opposite  tJie  greater  angle,  and  conversely. 

If,  in  the  spherical  triangle  AB  C,  the  angle  B  is  greater 
than  the  angle  (7,  it  is  to  be  proved  that  the  side  AC i^,  greater 
than  AB.  Making  the  angle  CBD  equal  to  (7,  we  have  DB  Cy 
an  isosceles  triangle.     Now, 

AC  =  AD-{-I>C  =  AI)^2)B>AB. 

Conversely,  the  greater  angle  is  opposite  the  greater  side. 

1*7.  Theorem. — Iii  a  right-angled  spherical  triangle,  one  of 
the  other  angles  and  its  opposite  side  are  either  both  equal  to 
ninety  degrees,  or  both  less,  or  both  greater. 

Suppose  A  the  right  angle,  and  0  the  center  of  the  sphere. 
Then,  1st,  if  the  side  AB  is  a  quadrant,  B  is  the  pole  of  the  arc 
A C,  for  the  radius  BO  is  perpendicular  to  the  plane  of  A C, 
Therefore  the  arc  BC  is  perpendicular  to  A  C,  that  is,  the  angle 
opposite  the  quadrant  AB  is  right. 

2d.  If  the  side  AJEJ  is  less  than  a 
quadrant,  £J  being  on  the  arc  AB  and  ^  B 

JE'jP  being  the  hypotenuse,  then  join  I^B.  / 

Now  BFA  is  a  right  angle,  but  EFA  is  / 

a  part  of  BFA.  / 

3d.  If  E  is  beyond  B,  that  is,  if  the  / 

side  is  greater  than  the  quadrant  AB,       ^  \I^ 
by  joining  the  vertex  of  the  opposite 


230  GEOMETRY  OF  SPACE.  [Chap.  XL 

angle  to  B  it  is  shown  that  that  angle  is  greater  than  a  right 
angle,  the  whole  being  greater  than  a  part. 

Two  angular  quantities  are  said  to  be  of  the  same  species 
when  both  are  less  than,  or  both  equal  to,  or  both  greater  than 
ninety  degrees. 

Polar  Triangles. 

18.  If,  at  the  vertex  of  a  triedral,  a  perpendicular  is  erected 
to  every  face,  these  lines  form  the  edges  of  a  supplementary 
triedral.  If  the  vertex  is  at  the  center  of  a  sphere,  there  are  two 
spherical  triangles  corresponding  to  these  triedrals,  and  they 
have  the  same  relations  as  two  supplementary  triedrals. 

Since  each  edge  of  one  triedral  is  perpendicular  to  the  oppo- 
site face  of  the  other,  the  vertex  of  each  angle  of  one  triangle  is 
the  pole  of  the  opposite  side  of  the  other.  Such  triangles  are 
called  polar  triangles,  and  sometimes  supplementary. 

Theorem. — If,  with  the  vertices  of  a  spherical  triangle  as 
poles,  three  arcs  of  great  circles  are  made,  a  second  triangle  is 
formed  whose  vertices  are  poles  of  the  sides  of  the  first. 

Let  ABC  he  the  given  triangle, 
and  EF,  BF,  and  BF  arcs  of  great 
circles  whose  poles  are  respectively  y^^,^^ 

A,  B,  and  G.  /    ^V\ 

Since  A  is  the  pole  of  the  arc  /        /  \      \ 

FF,  the  distance  from  J.  to  i^  is  /  /       \     \ 

one  quadrant,  and,  since  B  is  the  /  /     jj^.'''    \ 

pole  of  the  arc  BF,  the  distance       /;(:::"""  i'^'^^J..^-^^ 
from   B    to  F  IB    one    quadrant. 
Therefore  F  is  the  pole  of  the  arc 

AB.     In  the  same  way  it  is  proved  that  B  is  the  pole  of  the  arc 
B  C,  and  that  F  is  the  pole  of  the  arc  A  C. 

If  BFF  were  the  given  triangle,  the  polar  triangle  formed 
from  it  would  be  ABC\  each  is  the  polar  triangle  of  the 
other. 

19.  Theorem. —  Of  two  polar  triangles,  every  angle  of 
either  triangle  and  the  opposite  side  of  the  other  are  supple- 
mentary. 


Art.  19.]  SOLIDS  OF  REVOLUTION,  231 

In  the  two  triangles  AB  C  and  DEF^  the  pole  of  AB  is  2>, 
the  pole  oi  AC  IB  F,  etc.     The  sum  of  FB  and  GH  is  equal  to 
the  sum  of  the  two  quadrants 
FH  and  GB,  but  the  arc  GB[  p 

is  equal  in  angular  quantity  to  [\    B 

the    anffle  A.      Therefore   the  /   \^\ 

angle  A  and  the  arc  i^2>  are  /|       \      X 

supplementary.     By  producing  X    I         \  \(7 

the  side  BF  both  ways  to  meet        jr/  \        m^-^^  \ 

BA  at  ^  and  B  C  produced  at       ^^-^^^^.e.rjc^"'\^^  -^iV 

iVJ  it  may  be  shown  that  B  and  ^'         JJ 

BF  are  supplementary.     Simi- 
larly F  is  the  supplement  of  A  C,  etc. 

These  two  theorems  are  corollaries  of  the  theorem  on  sup- 
plementary triedrals.  Compare  the  two  modes  of  demonstra- 
tion. 

The  name  "  polar  "  refers  to  the  position  of  the  triangles  as 
described  in  the  first  theorem.  Two  supplementary  triangles  do 
nob  necessarily  have  any  relative  position.  Another  triangle 
equal  to  BFF  might  be  on  any  part  of  the  sphere,  and  such  a 
triangle  and  the  triangle  AB  C  would  have  the  reciprocal  rela- 
tion stated  in  this  second  theorem. 

The  student  may  derive  assistance  from  diagrams  on  a  globe. 
Draw  the  polar  triangle  of  a  birectangular  triangle,  of  a  trirect- 
angular  triangle,  of  a  very  small  triangle,  etc.,  etc. 

30.  Exercises. — 1.  A  plane  tangent  to  a  sphere  at  the  pole  of  a 
circle  is  parallel  to  the  plane  of  that  circle. 

2.  In  a  right-angled  spherical  triangle,  if  one  of  the  oblique  angles  is 
acute,  it  is  greater  in  angular  quantity  than  its  opposite  side ;  if  obtuse,  it 
is  less.  But,  when  two  angles  of  a  triangle  are  right,  the  third  angle  and 
its  opposite  side  are  equal  in  angular  quantity.  [Consult  the  diagram  of 
Article  15,  Corollary  IX.] 

3.  If  arcs  of  any  two  circles  cut  each  other,  their  angle  is  equal  to  the 
diedral  formed  by  two  planes,  each  of  which  passes  through  a  line  tan- 
gent to  one  of  the  curves  at  their  intersection,  and  through  the  center  of 
the  sphere. 

4.  If  arcs  on  a  sphere  are  tangent  to  the  same  straight  line,  they  are 
tangent  to  each  other. 


GEOMETRY  OF  SPACE.  [Chap.  XL 

Spherical  Areas. 

21.  Theorem. — The  area  of  the  curved  surface  of  an  in- 
scribed frustum  of  a  cone  is  equal  to  the  product  of  the  altitude 
of  the  frustum  by  the  circumference  of  a  circle  whose  radius  is 
the  perpendicular  let  fall  from  the  center  of  the  sphere  upon  the 
slant  height  of  the  frustum. 

Let  AEFJ)  be  the  semicircle  that  de- 
scribes the  given  sphere,  and  EBHF  the 
trapezoid  that  describes  the  frustum. 
Make  EK  perpendicular  to  FH.  Then 
EK  is  the  altitude  of  the  frustum.  Let 
IC  be  the  perpendicular  from  the  center 
of  the  sphere  on  EF.  Then  the  circum- 
ference of  a  circle  of  this  radius  is  2rcCI. 
It  is  to  be  proved  that  the  area  of  the 
curved  surface  of  the  frustum  is  equal  to 
the  product  of  2r:CIxEK. 

The  chord  EF  is  bisected  at  the  point 
I.  Make  GI  perpendicular  to  AD.  The  point  I  in  its  revolu- 
tion describes  the  circumference  of  the  section  midway  between 
the  two  bases  of  the  frustum.  GI  is  the  radius  of  this  circum- 
ference, which  is  therefore  2tt  GI.  The  area  of  the  curved  sur- 
face of  the  frustum  is  equal  to  the  product  of  the  slant  height 
by  this  circumference  ;  that  is,  ^ttGIxEF. 

The  triangles  EFK  and  IG  (7,  having  their  sides  respective- 
ly perpendicular  to  each  other,  are  similar.     Therefore, 

EF'.EK  =  CI:GL 
Therefore,  GI  X  EF  =CIX  EK,  and 

2t:GIxEF=  2irCIxEK. 

As  the  first  member  of  this  equation  is  equal  to  the  area  of 
the  curved  surface  of  the  frustum,  the  second  is  equal  to  the 
same  area. 

Corollary. — If  the  vertex  of  the  cone  were  at  the  point  A, 
the  cone  itself  would  be  inscribed  in  the  sphere.  The  curved 
surface  of  an  inscribed  cone  is  equal  to  the  product  of  its  alti- 
tude by  the  circumference  of  a  circle  whose  radius  is  a  perpen- 


Art.  21.]  SOLIDS  OF  REVOLUTION.  233 

dicular  let  fall  from  the  center  of  the  sphere  upon  the  slant 
height. 

23#  Theorem, — The  area  of  the  surface  of  a  sphere  is  equal 
to  the  product  of  the  diameter  by  the  circumference  of  a  great 
circle. 

Let  ADEFGB  be  the  semicircle  by 
which  the  sphere  is  described,  having  inscribed 
in  it  half  of  a  regular  polygon  which  revolves 
with  it  about  the  common  diameter  AB. 

Then,  the  surface  described  by  the  side 
AD  is  equal  to  2?:  GI  by  AH.      The  surface 
described  by  DE  is  equal  to  27r  GI  by  HK^ 
for  the  perpendicular  let  fall  upon  DE  is 
equal  to  GI\  and  so  on.     If  one  of  the  sides, 
as  EF^  is  parallel  to  the  axis,  the  measure  is 
the  same,  for  the  surface  is  cylindrical.     Add- 
ing these  equal  quantities,  the  sums  are  equal.     That  is,  the  en- 
tire surface  described  by  the  revolution  of  the  regular  polygon 
about  its  diameter  is  equal  to  the  product  of  the  circumference 
whose  radius  is  Glhj  the  diameter  AB. 

Let  the  polygon  vary,  doubling  the  number  of  sides  at  every 
step.  The  radius  of  the  sphere  is  the  limit  of  the  variable  apo- 
them  GI\  therefore  the  limit  of  the  circumference  having  this 
variable  radius  is  the  circumference  of  a  great  circle.  The  limit 
of  the  variable  perimeter  is  the  semi-circumference  ;  therefore 
the  limit  of  the  surface  described  by  the  perimeter  is  the  surface 
of  the  sphere.  Therefore  the  area  of  the  surface  of  a  sphere  is 
equal  to  the  product  of  the  diameter  by  the  cii'cumference  of  a 
great  circle. 

Corollaries. — I.  The  area  of  the  surface  of  a  sphere  is  four 
times  the  area  of  a  great  circle.  For  the  area  of  a  circle  is  equal 
to  the  product  of  its  circumference  by  one  fourth  of  the  diame- 
ter.    The  spherical  area  may  be  expressed  by  4:TtII^, 

II.  The  area  of  the  quadrantal  triangle  is  one  half  that  of  a 
great  circle.     It  may  be  expressed  by  ^ttB^, 

III.  The  area  of  a  zone  is  equal  to  the  product  of  its  altitude 
by  the  circumference  of  a  great  circle.     For  the  area  of  the 

11 


234 


GEOMETRY  OF  SPACE, 


[Chap.  XI. 


zone  described  by  the  arc  AD  is  equal  to  the  product  of  AH 
by  the  circumference  whose  radius  is  the  limit  of  the  variable 
apothem. 

TV,  The  area  of  the  surface  of  a 
sphere  is  equal  to  the  area  of  the  curved 
surface  of  a  circumscribing  cylinder. 

If  a  semicircle  is  inscribed  in  a  rect- 
angle and  the  figure  is  revolved  about 
the  diameter  as  an  axis,  a  sphere  and  a 
cylinder  are  generated,  and  the  sphere 
is  inscribed  in  the  cylinder  ;  that  is,  the 
curved  surface  and  both  bases  of  the 
cylinder  are  tangent  to  the  sphere. 
The  altitude  of  the  cylinder,  the  diam- 
eter of  its  base,  and  the  diameter  of  the  sphere  are  equal. 

V.  The  area  of  a  lune  is  to  the  area  of  the  whole  spherical 
surface  as  the  angle  of  the  lune  is  to 
four  right  angles. 

For  the  extent  of  the  lune  and  of  its 
surface  vary,  as  the  angle  D  GE^  or  its 
equal  DAE. 

Let  n  be  the  number  of  right  angles 
in  the  angle  of  the  lune,  then  the  area 

is  -  of  the  area  of  the  sphere,  and  it  is  ^ 

2n  times  the  area  of  the  quadrantal  triangle.  It  may  be  ex- 
pressed by  TTfili^. 

Areas  of  Triangles. 

!33.  Theorem. — Two  symmetrical  spherical  triangles  are 
equivalent. 

Let  AEI  and  B  CD  be  two  symmetrical  triangles,  the  angle 
A  being  equal  to  B,  E  to  C,  and  I  to  D.  Then  the  side  AE 
is  equal  to  BC,  AI  to  BD,  and  EI  to  CD,  but  the  triangles 
are  not  superposable.  It  is  to  be  proved  that  they  have  equal 
areas. 

Let  a  circle  be  described  about  each  triangle,  the  poles  being 
O  and  F.     These  two  circles  are  equal  (15,  xiii). 


Art.  23.] 


SOLIDS  OF  REVOLUTION. 


235 


By  arcs  of  great  circles  join  OA,  OE,  and  01  \  and,  in  the 
same  way,  join  FB,  FC,  and  FB. 

The  triangles  A  01  and  BFB 
are  isosceles,  and  mutually  equi- 
lateral ;  for  A  0, 10,  BF,  and  BF 
are  equal  arcs  (10,  ii).  Hence, 
these  triangles  are  equal  (15,  iii). 
For  a  similar  reason,  the  triangles 
I  OF  and  GFB  are  equal ;  also, 
the  triangles  A  OF  and  BFC. 
Therefore  the  areas  of  the  trian- 
gles AFI  and  BOB,  being  the 
differences  of  equals,  are  equal. 

The  pole  of  the  small  circle  may  be  inside  of  the  given  trian- 
gle, in  which  case  each  of  the  original  triangles  is  the  sum  of 
three  isosceles  triangles. 


34,  Theorem. — If  the  quadrantal  triangle  is  taJcen  as  the 
unit  of  spherical  area,  and  the  right  angle  as  the  unit  of  angu- 
lar quantity,  the  area  of  any  spherical  triangle  is  indicated  by 
its  spherical  excess. 

That  is,  if  there  are  a  right  angles  in  the  spherical  excess,  the 
area  is  a  times  that  of  the  quad- 
rantal triangle. 

Let  AB  G  be  any  spherical  tri- 
angle. Produce  the  sides  around 
the  sphere  and  consider  the  great 
circle  BGFB  as  the  plane  of  ref- 
erence. 

Let  m,  n,  and  p  represent  the 
number  of  right  angles  in  the  sev- 
eral angles  A,  B,  and  G.     Then 

m,  n,  and  p  may  each  represent  any  number  less  than  2.  The 
spherical  excess  is  {in-\-n-\-p  —  2)  right  angles,  and  it  is  to  be 
proved  that  the  area  of  the  triangle  is  {m -\- n -\- p  —  2)  times 
the  quadrantal  triangle. 

The  area  of  the  lune  AB  0  GA  is  2;?^  quadrantal  triangles. 
The  triangle  BOG,  which  is  a  part  of  this  lune,  is  equivalent  to 


236  GEOMETRY  OF  SPACE.  [Chap.  XL 

its  opposite  and  symmetrical  triangle  DAF.  Therefore  the  area 
of  the  two  triangles  ABC  d^ndi  DAF=  2m  quadrantal  triangles. 
The  area  of  the  lune  BCFAB  =  2n  quadrantal  triangles.  The 
area  of  the  lune  GBDA  C  ~2p  quadrantal  triangles. 

In  the  sum  of  these  three  lunes,  the  triangle  AB  C  is  included 
three  times,  and  the  rest  of  the  spherical  surface,  above  the  plane 
of  reference,  once.  The  sum  is  the  area  of  the  hemisphere,  plus 
twice  the  area  of  the  triangle  AB  C,  but  the  area  of  the  hemi- 
sphere is  equal  to  that  of  four  quadrantal  triangles.  Then,  the 
areas  of 

4  quad.  tri.  -j-  2  triangles  AB  C  =  2{m-{-n-{-p)  quad.  tri. 

Transposing  the  first  term  and  dividing  by  2, 

area  AB  C  =  {m-\-n-\-p  —  2)  quad.  tri. 

Corollaries. — I.  If  the  square  of  the  radius  is  the  unit  of 
area,  then  the  area  of  a  spherical  triangle  i%  ^  {m -\- n -{- p  —  2) 

II.  The  area  of  any  spherical  polygon  is  indicated  by  its 
spherical  excess. 

For  the  spherical  excess  of  the  polygon  is  the  sum  of  the 
spherical  excesses  of  the  ti'iangles  that  compose  it ;  and  its  area 
is  the  sum  of  their  areas. 

Triedrals  and  other  polyedrals  are  measured  by  their  spher- 
ical excess.  The  sum  of  the  diedral  angles  over  the  sum  of  the 
angles  of  a  plane  polygon  of  the  same  number  of  sides  indicates 
what  portion  of  space  is  included  between  the  faces.  Ordinarily 
it  is  called  the  spherical  area  for  a  sphere  having  unity  for  its 
radius. 

This  is  applied  in  measurements  of  anything  radiating  in  all 
directions  from  a  point,  such  as  light,  heat,  etc.  The  portion  in- 
tercepted by  any  body,  as  compared  with  the  whole  amount 
radiated,  is  accurately  measured  by  means  of  the  spherical  ex- 
cess ;  or  it  may  be  called  the  spherical  area  for  radius  unity. 


Art.  25.] 


SOLIDS  OF  REVOLUTION. 


237 


Volume  of  the  Sphere. 

35.  Theorem. — The  volume  of  a  sphere  is  equal  to  one  third 
of  the  product  of  the  surface  by  the  radius. 

Suppose  a  polyedron  circumscribed  about  the  sphere.  A 
plane  may  pass  through  each  edge  of  the  polyedron,  and  extend 
to  the  center  of  the  sphere.  These  planes  divide  the  polyedron 
into  as  many  pyramids  as  the  figure  has  faces.  The  faces  of  the 
polyedron  are  the  bases  of  the  pyramids. 

The  altitude  of  each  is  the  radius  of  the  sphere,  for  the  radius 
to  the  point  of  tangency  is  perpendicular  to  the  tangent  plane. 
The  volume  of  each  pyramid  is  one  third  of  its  base  by  its  alti- 
tude. Therefore,  the  volume  of  the  polyedron  is  one  third  the 
sum  of  the  bases  by  the  common  altitude,  or  radius. 

A  sphere  is  the  limit  of  a  variable  circumscribed  polyedron 
which  varies  according  to  a  certain  law.  This  may  be  demon- 
strated by  a  reasoning  like  that  in  the  analogous  case  of  the  cir- 
cle, showing  that  at  every  step  the  space  between  the  sphere  and 
the  variable  polyedron  is  diminished  by  more  than  a  given  frac- 
tional part  of  the  remainder.  The  following  demonstration  by 
the  method  of  infinites  is  more  elegant  and  equally  rigorous. 

Make  the  square  CADB  and  the  quadrant  BA,  having  its 
center  at  (7,  and  join  CD. 

If  this  figure  revolves 
upon  B  Cy  the  square  gener- 
ates a  cylinder,  the  quarter 
circle  BAG  generates  a 
hemisphere,  and  the  triangle 
BJDG  generates  a  cone.  This 
cylinder,  segment,  and  cone 
have  equal  bases,  that  is,  the 
circle  whose  radius  is  CA  or 
BJDy  and  they  have  the  same 
altitude,  BG. 

Through  any  point  of 
BG,  say  E,  make   a  plane 

parallel  to  the  base.     Three  circular  sections  are  made,  a  section 
of  the  sphere  having  the  radius  EH,  a  section  of  the  cylinder 


238 


GEOMETRY  OF  SPACE. 


[Chap.  XI. 


having  the  radius  EF,  and  a  section  of  the  cone  having  the 
radius  EG.     Join  GS.     Then,  since  CEH  is  a  right  angle, 

EH^=  CS^—EG^. 

But  GE:=  GA  =  EF,  and  EG  =  EG.  Substituting  and 
multiplying  by  tt, 

ttEE:^  =  ttEF^  —  ttEG^, 

that  is,  the  section  of  the  sphere  is  equivalent  to  the  difference 
of  the  sections  of  the  cylin- 
der and  of  the  cone. 

The  equation  between 
these  sections  is  true  when 
E  is  at  any  point  of  the  axis 
£G.  Suppose  a  plane  par- 
allel to  the  base  through 
every  point  of  £G.  The 
hemisphere  is  composed  of 
the  sum  of  this  infinite  num- 
ber of  parallel  spherical  sec- 
tions, that  is,  the  sum  of 
the  terms  represented  by 
ttEIT^.      The     cylinder     is 

composed  of  the  infinite  number  of  terms  represented  by 
ttEF^,  and  the  cone  of  those  represented  by  ttEG^.  Every 
term  in  the  first  of  these  three  infinite  series  has  its  correspond- 
ing term  in  the  second  series,  and  in  the  third.  Therefore  the 
same  equality  exists  between  the  sums  as  between  three  corre- 
sponding terms  ;  and  the  volume  of  the  hemisphere  is  equal  to 
the  difference  of  the  volumes  of  the  cylinder  and  the  cone. 

Since  the  volume  of  the  cylinder  is  equal  to  the  product  of 
the  base  by  the  altitude,  and  the  volume  of  the  cone  to  one  third 
of  that  product,  the  volume  of  the  hemisphere  is  equal  to  two 
thirds  of  the  product  of  a  great  circle  by  the  radius,  and  the  vol- 
ume of  a  sphere  is  equal  to  four  thirds  of  the  product  of  a  great 
circle  by  the  radius,  ^nH^.  The  area  of  the  surface  being  four 
times  that  of  a  great  circle,  the  volume  is  one  third  of  the  prod- 
uct of  the  surface  by  the  radius. 


Art.  25.]  SOLIDS  OF  REVOLUTION.  239 

Corollaries. — I.  The  volume  may  be  expressed  by  ^rrD^,  or 
by  iirUK 

II.  The  volume  of  a  spherical  pyramid,  or  of  a  spherical 
wedge,  or  of  a  spherical  sector,  is  equal  to  one  third  of  the  prod- 
uct of  the  area  of  its  spherical  surface  by  the  radius. 

The  volume  of  a  spherical  segment  of  one  base  is  equivalent 
to  the  sum  or  to  the  difference  of  the  volume  of  a  cone  and  that 
of  a  sector.     For  the  sector 
ABCI>  is  composed  of  the 
segment  ABC  and  the  cone 
A  CD. 

The  volume  of  a  spherical 
segment  of  two  bases  is  the 
difference  of  the  volumes  of 
two  segments  each  of  one 
base.  Thus  the  segment 
AEFG  is  equal  to  the  segment  ABC  less  EBF. 


2G»  Theorem. —  The  areas  of  the  surfaces  of  two  spheres 
are  to  each  other  as  the  squares  of  their  diameters;  and  their 
volumes  are  as  the  cubes  of  their  diameters^  or  other  homologous 
lines. 

Let  D  and  d  represent  the  diameters  of  any  two  spheres. 
Then  the  areas  are  7r7>2  and  'nd'^^  whose  ratio  is  J>2  .  <^2^  The 
volumes  are  ^-rxD^  and  ^ttc^^,  whose  ratio  is  D^  :  d^. 

Since  solids  with  curved  surfaces  are  the  limits  of  variable 
polyedrons,  any  similar  figures  have  for  their  superficial  ratio  the 
square  of  the  linear  ratio,  and  for  their  solid  ratio  the  cube  of 
the  linear. 

2*7.  Scholium. — Our  subject  has  developed  from  the  sim- 
plest linear  figures  to  the  doctrine  of  the  cone  and  the  sphere. 
The  topics  are  nearly  the  same  as  were  discussed  two  thousand 
years  ago  by  Euclid  and  Archimedes.  Euclid  arranged  the 
matter  more  for  convenience  of  demonstration  than  with  refer- 
ence to  a  classification  of  magnitudes  (II,  15  and  16).  His  first 
theorems  were  about  triangles.     From  these  the  properties  of 


240  GEOMETRY  OF  SPACE.  [Chap.  XI. 

other  inclosed  figures  and  of  angles  and  parallel  lines  were  de- 
duced. Until  within  a  century,  geometers  have  pursued  sub- 
stantially the  same  order.  Vincent  was  (I  believe)  the  first  to 
introduce,  about  1830,  the  arrangement  according  to  a  logical 
classification  of  magnitudes. 

This  logical  order  lets  the  student  see  that  there  are  fields  of 
geometrical  research  on  both  sides  of  the  path  he  pursues.  The 
science  of  geometry  is  not  merely  a  specimen  of  rigorous  demon- 
stration, although  some  famous  teachers  have  seen  nothing  else 
in  it. 

During  the  present  century  the  theory  of  proportional  lines 
has  been  very  much  developed.  Centers  of  similarity,  the  ratios 
arising  from  lines  meeting  at  one  point  and  cut  by  other  lines 
called  transversals,  and  many  theorems  more  or  less  related  to 
these  notions,  constitute  almost  a  distinct  science,  which  has  been 
called  Modern  Geometry. 

The  properties  of  the  Conic  Sections  have  been  developed  in 
ancient  times  by  the  Euclidean  method,  and  more  exhaustively 
in  modern  times  by  algebraic  methods.  The  application  of  al- 
gebra to  the  investigation  of  magnitudes  constitutes  the  science 
of  Analytic  Geometry. 

The  calculation  of  angles  and  of  distances  belongs  to  the 
science  of  Trigonometry.  Mensuration,  or  the  art  of  measuring, 
is  an  application  of  Geometry  and  Trigonometry. 

No  real  progress  can  be  made  in  these  studies  without  fre- 
quent use  of  the  elementary  principles  of  Geometry. 


38«  Exercises. — 1.  Find  the  area  of  the  earth's  surface,  suppos- 
ing it  to  bo  a  sphere  with  a  diameter  of  7,912  miles. 

2.  Find  what  part  of  the  surface  is  between  the  equator  and  the  par- 
allel of  30°  north  latitude. 

3.  Find  what  part  of  the  surface  is  between  two  meridians  which  are 
ten  degrees  apart. 

4.  Find  the  area  of  a  triangle  described  on  a  globe  of  13  inches  diam- 
eter, the  angles  being  100°,  45°,  and  53°. 

5.  Discuss  the  possible  relative  positions  of  two  spheres. 

6.  If  a  spherical  triangle  has  one  side  a  quadrant,  another  side  and  its 
opposite  angle  are  of  the  same  species. 


Abt.  28.] 


SOLIDS  OF  REVOLUTION, 


241 


7.  The  surface  of  a  sphere  can  bo  completely  covered  with  either  4, 
or  8,  or  20  equilateral  spherical  triangles. 

8.  The  volume  of  a  cone  is  equal  to  the  product  of  its  whole  sur- 
face by  one  third  the  radius  of  the  inscribed  sphere. 

9.  If,  about  a  sphere,  a  cylinder  is  circumscribed,  also  a  cone  whose 
slant  height  is  equal  to  the  diameter  of  its  base,  the  area  and  the  volume 
of  the  sphere  are  two  thirds  of  the  area  and  the  volume  of  the  cylinder ; 
and  the  area  and  the  volume  of  the  cylinder  are  two  thirds  of  the  area 
and  the  volume  of  the  cone. 


ee  y- 


NOTES. 


NOTE    A. 
General  Axioms,  Chapter  I,  Article  2. 

The  attempts  of  logicians  and  geometers  to  state  the  first  principles 
of  inference,  or  laws  of  thought,  have  not  resulted  in  concurrence  of 
opinion.  In  the  Elements  of  Euclid  there  are  nine  ' '  common  notions  " 
as  he  called  them.  Legendre  retained  two  of  these:  1.  Two  quantities 
equal  to  a  third  are  equal  to  each  other ;  and  2.  The  whole  is  greater  than 
its  part.  Six  of  Euclid's  axioms  have  been  summed  up  in  these  two :  1. 
If  the  same  operation  is  performed  on  equal  quantities  the  results  are 
equal,  and  2.  If  the  same  operation  is  performed  on  unequal  quantities 
the  results  are  unequal.  "With  these  Euclid  stated  an  axiom,  which  is 
geometrical,  and  which  Legendre  has  also  retained,  asserting  the  equiva- 
lence of  congruent  magnitudes.  Logicians  have  differed  as  to  whether 
this  is  merely  a  definition. 

The  above  apply  only  to  relations  of  extent ;  which  vary  by  addition, 
subtraction,  multiplication,  or  division.  An  axiom  that  covers  the  whole 
truth  should  include  changes  in  form,  or  in  any  other  attribute  of  things. 
For  example :  If  two  things  that  agree  in  a  common  attribute  are  changed 
in  the  same  way  and  to  the  same  amount,  the  results  must  agree  as  to  that 
attribute  ju^t  as  the  two  things  agreed.  Any  statement  of  the  axioms  or 
primary  laws  of  reasoning  is  omitted  from  the  text,  as  they  pertain  rather 
to  metaphysics  than  to  geometry,  and  no  statement  yet  made  seems  suf- 
cient. 


244  NOTES, 


NOTE    B. 


Postulates  and  Problems,  Chapter  I,   Article  2  ;    Chapter  II, 
Articles  3,  8,  13,  etc. 

It  was  taken  for  granted  by  Euclid  that  a  straight  line  may  have  any 
position,  and  that  its  length  may  be  increased  to  any  extent,  also  that  a 
circumference  of  a  circle  may  have  any  position  and  any  extent  (IV,  34). 
It  does  not  appear  that  he  regarded  these  as  self-evident  truths,  though 
in  modern  times  they  have  received  that  character.  They  are  only  a  j)ar- 
tial  statement  of  the  truth,  for  a  line  may  be  of  any  extent,  small  or  great ; 
so  may  any  magnitude;  and  a  magnitude  may  be  of  any  form.  All 
this  truth  is  as  indemonstrable  and  as  universal  as  the  part  demanded  by 
Euclid.  He  carefully  refrained  from  stating  anything  more  than  he 
thought  was  necessary  to  explain  the  construction  of  the  figm-es  con- 
tained in  his  Elements. 

Legendre  assumed  tacitly  the  possibility  of  every  figure  that  can  be 
precisely  defined.  He  stated  no  postulates,  but  made  any  construction 
that  was  not  self-contradictory.  This  method  removes  all  occasion  for 
problems  in  the  theory  of  pure  geometry.  Accordingly,  in  Legendre's 
work,  the  problems  are  all  problems  in  drawing.  These  are  relegated  to 
a  separate  place,  as  an  application  of  geometric  principles  to  an  art. 

Clearly,  Euclid  did  not  regard  his  problems  in  that  way.  He  pro- 
ceeded upon  the  rule  that  the  possibility  of  every  figure  must  be  demon- 
strated. All  that  was  deemed  necessary  to  show  this  of  the  figures  with- 
in the  scope  of  his  work  was  stated  in  his  postulates  concerning  the 
straight  line  and  the  circle.  He  therefore  limited  his  demands  to  these. 
The  rule  and  compasses  are  used  to  draw  straight  lines  and  circles,  and 
(long  after  Euclid's  day)  the  problems  came  to  be  associated  with  these 
instruments,  and  hence  to  be  regarded  as  problems  in  drawing.  But  the 
postulates  of  Euclid  demand  less  than  can  be  done  with  these  instru- 
ments in  some  respects,  and  more  in  others.  We  may  infer  that  his  pos- 
tulates and  the  problems  which  depend  on  them  were  intended  as  a  part 
of  pure  geometry.  They  stated  what  is  possible  with  geometrical  lines, 
which  are  not  visible  to  the  material  eye,  nor  can  be  made  with  rule  and 
compasses. 

There  are  two  reasons  for  stating  the  Postulates  of  Form  and  of  Ex- 
tent :  1.  Every  premise  should  be  expressed.  This  is  of  no  less  importance 
if  the  premise  is  assumed  as  a  first  principle,  for  the  student  ought  to  see 
and  examine  the  grounds  of  the  argument.  2.  The  statement  should  be 
as  broad  as  the  nature  of  the  truth;  for  the  student  may  infer  that  the 


NOTES.  245 

statement  made  is  the  whole  truth.  This  has  in  fact  occasioned  general 
error  as  to  the  character  of  certain  problems.  That  a  square  can  be 
equivalent  to  a  circle  has  been,  by  the  half -learned,  regarded  as  impos- 
sible, because  the  drawing  is  beyond  the  power  of  rule  and  compasses. 

Professor  De  Morgan,  of  University  College,  London,  among  his  fine 
criticisms  on  Euclid,  suggests  the  following  as  a  postulate  used  in  de- 
monstration by  superposition :  "Any  figure  maybe  removed  from  place 
to  place  without  alteration  of  form,  and  a  plane  figure  may  be  turned 
round  on  the  plane."  Demonstration  by  superposition  does  assume  this. 
It  is,  however,  a  corollary  of  the  postulates  in  the  text.  If  a  figure  can 
be  of  any  form  or  extent,  then  a  figure  may  be  the  result  of  combining  in 
any  way  two  that  are  given. 

The  proposed  postulate  that  **a  magnitude  may  have  any  position  "  / 
is  also  a  corollary.     Position  is  relative ;  the  position  of  anything  is  its  / 
direction  and  distance    from  another  thing  to  which  it  is  referred.  / 
Therefore  the  notion  of  a  position  involves  two  things  in  one  geometricali 
figure.     The  two  are  parts  of  one  magnitude  or  combination,  and  their  \ 
relative  position  is  one  element  of  the  form  of  that  total.     Tliis  so-called 
"postulate  of  position"  is  therefore  included  in  the  Postulate  of  Form. 

Form  and  extent  are  the  only  properties  of  magnitude.  Therefore  these 
two  postulates  assert  the  possible  existence  of  whatever  can  be  thought 
of  magnitudes. 

NOTE    C. 
"Direction"  in  Definition,  Chapter  II,  Articles  6,  7,  10,  and  14. 

Direction  is  as  certain  and  definite  a  term  as  distance.  Both  words 
are  used,  without  ambiguity,  by  every  mathematician.  No  one  attempts 
a  definition  of  either.  Two  directions  may  be  identical,  and  two  distances 
may  be  equal.  If  the  distances  vary  equally  and  in  the  same  way,  the  re- 
sults are  equal.  If  the  directions  vary  equally  and  in  the  same  way,  the 
results  are  identical. 

Objectors  to  the  use  of  direction  in  defining  angle  and  parallel  admit 
the  propriety  of  saying  the  direction  of  a  straight  line  is  uniform — that 
the  straight  line  AB  may  be  produced  "in  the  same  direction"  to  G. 
Now,  if,  beyond  B,  there  can  be  one  line  and  only  one  in  the  direction 
AB^  it  is  equally  true  that  from  i>,  a  point  to  one  side,  there  can  be  a 
straight  line  having  the  same  direction  AB.  Not  only  is  this  possible, 
but  the  idea  is  more  simple  and  precise  than  any  other  definition  of  par- 
allels. The  possibility  is  an  immediate  inference  from  the  Postulate  of 
Form. 


246  NOTES, 

In  a  finite  straight  line  there  are  two  elements,  one  of  extent  and  one 
of  form.  For  purposes  of  definition,  Archimedes  used  the  element  of  ex- 
tent: *^A  straight  line  is  that  which  is  the  shortest  from  one  point  to  an- 
other." In  this  he  was  followed  by  Legendre  and  other  eminent  modern 
geometers.  Euclid  used  the  element  of  form :  "  A  straight  line  is  one  that 
lies  evenly  as  to  its  points."  This  is  better,  for  the  essential  character  of 
a  geometrical  figure  is  its  form.  But  this  Euclidean  definition  was  ob- 
scure and  barren ;  it  was  not  used  in  the  Elements.  The  Greeks  made  the 
same  word  serve  for  "straight "  and  for  "direction."  The  second  postu- 
late has  evdeiav  iTr*  evdtiag  kK^dlleiv. 

Euclid's  definition  of  a  "plane  angle"  was  "the  leaning,  K)daiQ^  to 
each  other  of  two  lines  that  meet  and  are  in  one  plane  and  not  in  the 
same  direction."  "When  the  lines  are  straight,  the  angle  is  called  recti- 
linear." Here  is  a  near  approach  to  the  simplest  idea  of  angular  quan- 
tity. The  ancient  geometer  saw  that  two  lines  must  be  "not  in  the 
same  direction  "  from  the  point  of  meeting,  but  he  did  not  see  that  two 
separate  lines  may  be  in  the  same  direction.  Hence  liis  definition  of  par- 
allels (III,  19)  fails  to  show  the  true  relation  between  angles  and  paral- 
lels.    This  is  done  only  when  they  are  seen  to  be  species  of  one  genus. 

Early  in  the  eighteenth  century,  the  German  philosopher,  Wolff,  pro- 
posed to  define  parallels  as  straight  lines  that  are  everywhere  equidistant, 
in  which  he  has  been  followed  by  several  eminent  authors.  The  use  of 
distance  is  as  objectionable  in  the  definition  of  parallels  as  in  that  of 
straight  lines.  It  belongs  with  that  definition  of  angles  which  makes 
them  contain  a  portion  of  the  indefinite  or  infinite  plane. 

De  Morgan,  the  most  rigorous  logician  of  the  English  mathematicians 
of  the  present  century,  said  ' '  permanency  of  direction  and  straightness 
are  equivalent  notions  "  (Penny  CyclopoBdia,  Direction).  In  another  ar- 
ticle, he  said,  "also  the  notion  of  differing  directions  is  suggested  by  two 
lines  which  make  an  angle";  and  he  added,  "we  may  readily  see  that 
the  relation  of  situation  which,  adopting  Euclid's  term,  may  be  called 
parallelism,  is  really  that  which  would  be  also  conveyed  by  the  words 
sameness  of  direction."  This  was  in  1838.  He  does  not  seem  to  have 
known  that  straightness  and  parallelism  had  been  defined  in  this  way, 
nine  years  previously,  by  an  American  author,  Hayward. 

The  English  "Association  for  the  Improvement  of  Geometrical 
Teaching  "  declares  that  an  angle  is  incapable  of  definition,  but  says  that 
"the  angle  is  greater  as  the  quantity  of  turning  is  greater  "  when  one  arm 
turns  about  the  vertex.  The  idea  of  rotation  is  good  for  the  explanation 
of  variable  angles,  which  become  greater  than  a  straight  angle.  The 
turning  or  rotation  is  a  change  of  direction,  a  variation  of  angular  quan- 
tity.    But  an  angle  may  be  a  constant.     The  variation  of  quantity  and 


NOTES,  247 

the  rotation  of  an  arm  are  not  essential  characteristics.  When  divested 
of  these  accidental  attributes,  an  angle  is  merely  the  difference  of  two 
directions.  When  this  difference  is  zero,  that  is  when  the  directions  are 
identical,  the  result  is  either  parallelism  or  coincidence. 


NOTE    D. 

Axioms  of  Direction,  Chapter  II,  Article  7. 

A  statement  of  the  principles  that  have  been  assumed  by  geometers 
to  be  primary  truths  concerning  angles  and  parallels  may  assist  the  stu- 
dent to  judge  correctly  what  are  the  fundamental  principles  of  direction. 

Euclid  said,  in  addition  to  the  three  postulates  in  the  text  (IV,  33), 
**  Let  it  be  granted 

*'4.  That  all  right  angles  are  equal ; 

*'5.  That  if  a  straight  line  meeting  two  straight  lines  makes  the  in- 
terior angles  on  the  same  side  less  than  two  right  angles,  the  two  straight 
lines  being  produced  infinitely  meet  each  other  on  that  side  on  which  the 
angles  are  less  than  two  right  angles ;  and 

*'  6.  That  two  straight  lines  do  not  enclose  a  space." 

Legendre  expressed  no  postulates  and  only  one  geometrical  axiom: 
"From  one  point  to  another  only  one  straight  line  can  be  drawn."  This 
is  generally  followed  by  geometers  on  the  Continent. 

In  England  editors  of  Euclid  have  proposed  various  substitutes  for 
the  fifth  demand. 

Simson's  Axiom. — *'A  straight  line  can  not  first  come  nearer  to 
another  straight  line,  and  then  go  further  from  it,  before  it  cuts  it ;  and, 
in  like  manner,  a  straight  line  can  not  go  further  from  another  straight 
line,  and  then  come  nearer  to  it ;  nor  can  a  straight  line  keep  the  same 
distance  from  another  straight  line,  and  then  come  nearer  to  it,  or  go 
further  from  it;  for  a  straight  line  keeps  always  the  same  direction." 

Playf air's  Axiom. — ''Two  straight  lines  which  intersect  one  another 
can  not  be  both  parallel  to  the  same  straight  line."  This  author  makes 
the  following  definition,  which  takes  the  place  of  Euclid's  sixth  postu- 
late :  "  If  two  lines  are  such  that  they  can  not  coincide  in  any  two  points, 
without  coinciding  altogether,  each  of  them  is  called  a  straight  line." 

Among  American  authors,  Hayward  has  two  axioms — the  one  given 
by  Legendre,  and  the  axiom  of  distance. 

Peirce  gives  the  axiom  of  distance  and  this:  ''The  direction  of  any 
point  of  a  straight  line  from  any  preceding  point  is  the  same  as  the 
direction  of  the  line  itself. " 


248  NOTSS. 

Loomis  states  two  axioms,  those  of  Playfair  and  of  Legendre,  and  he 
follows  Archimedes  in  defining  a  straight  line. 


NOTE    E. 

Some  Teems  variously  used. 

Equal  and  Equivalent,  Chapter  II,  Articles  3  and  4. — ^English  au- 
thors, following  Euclid  or  his  translators,  call  figures  that  have  the  same 
extent  "equal."  Legendre  used  the  word  "equivalent,"  and  this  has 
been  generally  followed  in  America,  and  on  the  Continent  of  Europe. 
In  order  to  express  what  we  call  "equal,"  the  Association  for  the  Im- 
provement of  Geometrical  Teaching  says  "identically  equal."  The  terms 
"congruent,"  and  "equal  in  all  respects,"  have  been  used  with  the  same 
meaning. 

Figure,  Chapter  II,  Article  5. — ^Euclid's  definition  of  axvi^o-^  figure, 
included  lines,  surfaces,  and  solids  of  limited  extent,  whatever  is  con- 
tained by  one  or  more  boundaries  or  ends.  Legendre,  whose  first  edition 
of  the  Elements  was  published  in  1794,  and  Playfair,  whose  first  edition 
of  Euclid  was  published  about  the  same  time,  both  applied  the  word 
figure  only  to  limited  plane  surfaces,  and  to  inclosed  portions  of  space. 
This  restricted  use  of  the  word  was  generally  adopted  till  within  the  last 
twenty  years.  In  1864,  seeing  that  a  term  was  needed  to  designate  "any 
magnitude  or  combination  of  magnitudes,"  I  made  this  the  definition  of 
figure.     It  has  been  adopted  by  several  authors. 

Polygon,  Chapter  II,  Article  15. — Euclid  did  not  include  triangles  or 
quadrilaterals  in  this  term.  Legendre  applied  it  to  all  plane,  rectilinear, 
inclosed  figures.  There  should  be  one  word  to  designate  this  class  of  ob- 
jects. 

Trapezoid,  Chapter  VI,  Article  1. — In  American  geometries  and  dic- 
tionaries, a  quadrilateral  with  two  sides  parallel  is  called  a  trapezoid,  but 
in  England  this  figure  is  generally  called  a  trapezium. 


INDEX. 


[This  Index  includes  the  terms  defined  in  the  text,  and  a  few  other  words.] 


Acute  angle,  iii,  6. 
Acute-angled  triangle,  v,  1. 
Adjacent  angles,  iii,  6. 
Adjacent  angles  in  a  triangle,  v,  1. 
Algebraic  method,  vi,  18. 
Alternate  angles,  iii,  20. 
Altitude  of  a  triangle,  v,  1. 

of  a  quadrilateral,  vi,  1. 

of  a  tetraedron,  x,  2. 

of  a  pyramid,  x,  5. 

of  a  prism,  x,  6. 

of  a  cone,  xi,  1. 

of  a  cylinder,  xi,  1. 

of  a  spherical  zone  and  segment,  xi,  14. 
Ancient  method,  vi,  18. 
Angle,  ii,  14. 

in  an  arc,  iv,  22. 

of  a  line  and  plane,  ix,  10. 

diedral,  ix,  13. 

spherical,  xi,  12. 
Apothem,  vii,  16. 
Arc,  iv,  3. 

containing  angle,  iv,  22. 
Area,  ii,  2. 

Arms  of  an  angle,  ii,  14. 
Axiom,  i,  2. 

of  Direction,  ii,  Y. 

of  Distance,  ii,  7. 
Axis,  ii,  9, 

of  Symmetry,  iv,  5. 

of  a  Circle,  ix,  8. 
Axles  of  wheels,  iv,  8. 


Base  of  a  triangle,  v,  1. 

of  a  quadrilateral,  vi,  1. 

of  a  tetraedron,  x,  2. 

of  a  pyramid,  x,  5. 

of  a  prism,  x,  6. 

of  a  cone,  xi,  1. 

of  a  cylinder,  xi,  1. 

of  a  spherical  segment,  xi,  14. 

of  a  spherical  pyramid,  xi,  14. 
Birectangular  triedral,  ix,  29. 
Broken  line,  ii,  6. 

Center  of  a  circle,  ii,  15. 

of  symmetry,  iv,  5. 

of  similarity,  v,  28. 

of  a  regular  polygon,  vii,  15. 

of  a  regular  polyedron,  x,  11. 

of  a  sphere,  xi,  1. 
Central  line,  iv,  29. 
Centroid  of  a  triangle,  x,  37. 
Chord,  iv,  3. 
Circle,  ii,  15. 

great,  xi,  8. 

small,  xi,  8. 
Circumference,  ii,  15. 
Circumscribed  circle  and  polygon,  v, 

cone  and  pyramid,  xi,  6. 

cylinder  and  prism,  xi,  7. 

sphere  and  polyedron,  xi,  8. 
Commensurable  lines,  iii,  1. 
Compasses,  iv,  33. 
Complementary,  iii,  6. 


250 


INDEX. 


Complements  of  angles,  iii,  6. 

of  parallelograms,  vii,  14. 
Concave  line,  iii,  3. 

polygon,  vii,  1. 

polyedral,  ix,  40. 
Concentric  circles,  iv,  1. 
Cone,  xi,  1. 
Conic  sections,  xi,  4. 
Conjugate  angles,  iii,  6. 

arcs,  iv,  9. 
Constant,  iii,  28. 
Construction,  ii,  5. 
Contact,  point  of,  iv,  16. 
Contained,  angle  by  arc,  iv,  22. 

rectangle  by  lines,  vi,  10. 
Contrapositive,  iii,  29. 
Converse  propositions,  i,  2. 
Convex  line,  iii,  3. 

polygon,  vii,  1. 

polyedral,  ix,  40. 
Corollary,  i,  2. 

Corresponding  angles,  iii,  20. 
Cube,  X,  6. 
Curvature,  ii,  6. 
Curve,  ii,  6. 
Curved  surface,  ii,  10. 
Cut,  line  by  plane,  ix,  3. 
Cylinder,  xi,  1. 


Edge  of  a  diedral,  ix,  13. 

of  a  triedral,  ix,  29. 

of  a  polyedron,  x,  1. 
Elements  of  a  triangle,  v,  9. 

of  a  triedral,  ix,  29. 
Equal  and  equality,  ii,  4. 
Equilateral  triangle,  v,  1. 

triedral,  ix,  29. 
Equivalent,  ii,  3. 
Escribed  circle,  v,  2. 
Excess,  spherical,  xi,  14. 
Exhaustions,  method  of,  iv,  31. 
Extent,  Postulate  of,  ii,  3. 
Exterior  angles,  iii,  20. 
External  center  of  similarity,  v,  28. 
Externally,  line  divided,  iii,  1. 
Extreme  and  mean  ratio,  iii,  2. 

Face  of  a  diedral,  ix,  13. 

of  a  triedral,  ix,  29. 

of  a  polyedron,  x,  1. 
Figure,  geometrical,  ii,  5. 

plane,  ii,  15. 
Foot  of  line  in  a  plane,  ix,  3. 
Form,  Postulate  of,  ii,  3. 

symmetry  of,  ix,  34. 
Frustum  of  a  pyramid,  x,  5. 

of  a  cone,  xi,  4. 


Decagon,  ii,  15. 
Determinate  problem,  iv,  34. 
Diagonal  of  a  polygon,  vi,  1. 

of  a  polyedron,  x,  1. 

plane,  ix,  40 ;  and  x,  1. 
Diameter,  iv,  1 ;  and  xi,  1. 
Diedral,  ix,  13. 
Direct  demonstration,  iv,  31. 

superposition,  iii,  31. 
Direction,  Axiom  of,  ii,  Y. 
Distance,  Axiom  of,  ii,  7. 
Division  of  lines,  iii,  1. 
Dodecaedron,  regular,  x,  11. 
Dodecagon,  ii,  15. 
Duplicate  ratio,  vi,  13. 


Gauge,  iii,  30. 
Geometry,  ii,  2. 
Given,  ii,  5  ;  and  iii,  1. 
Great  circle,  xi,  8. 

Harmonic  division,  iii,  2. 
Height,  slant  of  a  pyramid,  x,  5. 

of  a  frustum,  x,  5. 

of  a  cone,  xi,  1. 
Hemisphere,  xi,  1. 
Hexaedron,  regular,  x,  11. 
Hexagon,  ii,  15. 
Homologous,  ii,  3. 
Horizontal,  ix,  17. 
Hypotenuse,  v,  1. 


INDEX. 


251 


Icosaedron,  regular,  x,  11. 
In,  angle  in  arc,  iv,  22. 
Incommensurable  lines,  iii,  1  and  2. 
Indeterminate  problem,  iv,  34. 
Indirect  demonstration,  iv,  31. 
Indivisibles,  method  of,  iv,  31. 
Infinitesimals,  method  of,  iv,  31. 
Inscribed  angle,  iv,  22. 

circle  and  polygon,  v,  2. 

cone  and  pyramid,  xi,  6. 

cylinder  and  prism,  xi,  T. 

sphere  and  polyedron,  xi,  8. 

frustum  of  cone,  xi,  8. 
Interior  angles,  iii,  20. 
Internal  center  of  similarity,  v,  28. 
Internally,  line  divided,  iii,  1. 
Inverse  superposition,  iii,  31. 
Isoperimetrical,  vii,  20. 
Isosceles  triangle,  v,  1. 

triedral,  ix,  29. 

Lateral  edges  of  a  pyramid,  x,  5. 

of  a  prism,  x,  6. 
Length,  ii,  2. 
Light,  its  path,  vii,  26. 
Limit,  iii,  28. 
Limits,  method  of,  iv,  31. 
Line,  ii,  2. 

of  projection,  iii,  12. 
Linear  ratio,  v,  22. 
Locus,  iii,  12. 
Lozenge,  vi,  1. 
Lune,  xi,  14. 

Magnitude,  ii,  2. 

Major  arc,  iv,  9. 

Maximum,  vii,  20. 

Mean,  and  extreme,  ratio,  iii,  2. 

Medial,  v,  1. 

Method  of  superposition,  ii,  4 ;  iii,  31. 

of  exhaustions,  iv,  31. 

of  limits,  iv,  31. 

of  indivisibles,  iv,  31. 

of  infinitesimals,  iv,  31. 


Method,  for  exercises  in  drawing,  iv,  34. 

algebraic,  vi,  18. 

ancient,  vi,  18. 
Minimum,  vii,  20. 
Minor  arc,  iv,  9. 
Modes  of  reasoning,  iv,  31. 

Normal,  iv,  17. 

Oblique  lines  and  angles,  iii,  6. 
Obtuse  angle,  iii,  6. 
Obtuse-angled  triangle,  v,  1. 
Obverse,  iii,  31. 
Octaedron,  regular,  x,  11. 
Octagon,  ii,  15. 

Opposite  angles  in  a  triangle,  v,  1. 
Orthogonal  projection,  iii,  12. 

Parallel  lines,  ii,  14. 

(Euclid's  definition,  iii,  19). 

rulers,  vi,  7. 

line  and  plane,  ix,  19. 

planes,  ix,  20. 
Parallelism,  ix,  19. 
Parallelogram,  vi,  1. 
Parallelopiped,  x,  6. 
Pass,  plane  through  line,  ix,  3. 
Pentagon,  ii,  15. 
Pentedecagon,  ii,  15. 
Perimeter,  ii,  16. 
Perpendicular  lines,  iii,  6. 

line  and  plane,  ix,  6. 
Perspective,  ix,  27. 
Pierce,  line,  plane,  ix,  3. 
Plane,  ii,  10. 

(Euclid's  definition,  ii,  12). 

figures,  ii,  15. 

Geometry,  ii,  15. 

of  symmetry,  ix,  34. 

section,  ix,  40. 
Point,  ii,  2. 

of  contact,  iv,  16. 
Polar  radius,  xi,  10. 

triangles,  xi,  18. 


252 


INDEX. 


Poles,  xi,  8. 
Polyedral,  ix,  40. 
Polycdron,  x,  1. 
Polygon,  ii,  15. 

spherical,  xi,  14. 
Position,  symmetry  of,  ix,  34. 
Postulate,  i,  2. 

of  Form  and  of  Extent,  ii,  3. 

of  Euclid,  iv,  33. 
Practical  propositions,  i,  2. 
Prism,  X,  6. 
Problem,  i,  2. 

in  drawing,  iv,  33. 
Projection,  iii,  12. 

of  line  on  plane,  ix,  9. 
Protractor,  iv,  2*7. 
Ptolemaic  Theorem,  vi,  26. 
Pyramid,  x,  6. 

spherical,  xi,  14. 
Pythagorean  Theorem,  vi,  28. 

Quadrant,  iv,  20. 
Quadrantal  triangle,  xi,  15. 
Quadrature,  vi,  32. 

of  the  circle,  viii,  1 1. 
Quadrilateral,  ii,  15. 

Eadius  of  a  circle,  iv,  1. 

of  a  regular  polygon,  vii,  15. 

of  a  sphere,  xi,  1. 

polar,  xi,  10. 
Railway  curve,  iv,  27. 
Ratio,  extreme  and  mean,  iii,  2. 
Rectangle,  vi,  1. 
Rectangular  triedral,  ix,  29. 

parallclopiped,  x,  6. 
Rectification  of  a  curve,  viii,  8. 
Reductio  ad  absurdum^  iv,  31. 
Reflex  angle,  iii,  6. 
Regular  polygon,  vii,  1. 

pyramid,  x,  5. 

prism,  X,  6. 

polyedron,  x,  11. 
Reverse  face,  iii,  31. 


Revolution,  solid  of,  xi,  1. 
Rhombus,  vi,  1. 
Right  angle,  iii,  6. 

prism,  X,  6. 

solid,  X,  6. 
Right-angled  triangle,  v,  1. 
Ruler,  iv,  83 ;  and  vi,  7. 


Scalene  triangle,  v,  1. 
Scholium,  i,  2. 
Secant,  iii,  1. 
Section,  plane,  ix,  40. 
Sector  of  a  circle,  viii,  1. 

of  a  sphere,  xi,  14. 
Segment  of  a  line,  iii,  1. 

of  a  circle,  viii,  1. 

of  a  sphere,  xi,  14. 
Side  of  a  polygon,  ii,  15. 

of  a  tetraedron,  x,  2. 

of  a  pyramid,  x,  5. 

of  a  prism,  x,  6. 
Similar,  ii,  3  ;  and  ii,  14. 

ancient  definition,  v,  19. 
Similarity,  center  of,  v,  28. 
Slant  height  of  a  pyramid,  x,  5, 

of  a  frustum,  x,  5. 

of  a  cone,  xi,  1. 
Small  circle,  xi,  9. 
Solid,  ii,  2. 

of  revolution,  xi,  1. 
Space,  Geometry  of,  ii,  16. 
Species,  angles  of  same,  xi,  17. 
Sphere,  xi,  1. 
Spherical  angle,  xi,  12. 

excess,  xi,  14. 
Square,  vi,  1. 

an  instrument,  iii,  15  and  30. 
Squaring,  vi,  32. 

of  the  circle,  viii,  11. 
Stand,  angle  upon  arc,  iv,  22. 
Straight  line,  ii,  6. 

(definition  of  Euclid,  ii,  9). 

(definition  of  Archimedes,  ii,  9.) 

angle,  iii,  6. 


INDEX. 


253 


Subtend,  iv,  9. 

Superposition,  ii,  4 ;  iii,  31 ;  and  v,  13. 
Supplement  of  angle,  iii,  6. 
Supplementary  angles,  iii,  6. 

triedrals,  ix,  29. 

spherical  triangles,  xi,  18. 
Surface,  ii,  2. 
Surveying  land,  vi,  16. 
Symmetrical  figure,  iv,  5. 

points,  iv,  5. 

triedrals,  ix,  34. 
Symmetrically  similar,  x,  19. 
Symmetry,  center  and  axis  of,  iv,  5. 

of  form  and  of  position,  ix,  34. 

plane  of,  ix,  34. 

Tangent  lines,  iv,  16. 

surface,  xi,  1. 
Tetraedron,  x,  2. 
Theorem,  i,  2. 

Theoretical  proposition,  i,  2. 
Trapezoid,  vi,  1. 
Triangle,  ii,  15. 

an  instrument,  iii,  30. 


Triangle,  spherical,  xi,  15. 

quadrantal,  xi,  15. 
Triedral,  ix,  30. 
Trirectangular  triedral,  ix,  29. 
T-square,  iii,  30. 
Truncated  pyramid,  x,  5. 

Upper  base  of  frustum,  x,  5. 

Variable,  iii,  28. 

Vertex  of  an  angle,  ii,  14. 

of  a  triangle,  v,  1. 

of  a  triedral,  ix,  29. 

of  a  polyedron,  x,  1. 

of  a  tetraedron,  x,  2. 

of  a  pyramid,  x,  5. 

of  a  cone,  xi,  1. 
Vertical  angles,  iii,  6. 

line,  plumb,  ix,  1*7. 
Volume,  ii,  2. 

Wedge,  ix,  14. 

Zone,  xi,  14. 


THE    END. 


MATHEMATICS. 


Gillespie's  Land-Surveying.  Comprising  the  Theory- 
developed  from  Five  Elementary  Principles ;  and  the  Practice  with 
the  Chain  alone,  the  Compass,  the  Transit,  the  Theodolite,  the  Plane 
Table,  etc.  Illustrated  by  400  Engravings  and  a  Magnetic  Chart. 
By  W.  M.  Gillespie,  LL.  D.,  Civil  Engineer,  Professor  of  Civil  Engi- 
neering in  Union  College.     1  vol.,  8vo.    608  pages. 

A  double  object  has  been  kept  in  view  in  the  preparation  of  the  volume,  viz., 
to  make  an  introductory  treatise  easy  to  be  mastered  by  the  young  scholar  or  the 
practical  man  of  little  previous  acquirement,  the  only  prerequisites  being  arithme- 
tic and  a  little  geometry  ;  and,  at  the  same  time,  to  make  the  instruction  of  such  a 
character  as  to  lay  a  foundation  broad  enough  and  deep  enough  for  the  most  com- 
plete superstructure  which  the  professional  student  may  subsequently  wish  to  raise 
upon  it. 

Gillespie's  Higher  Surveying*.     Edited  by  Cady 

Staley,  a.  M.,  C.  E.  Comprising  Direct  Leveling,  Indirect  or  Trig- 
onometric Leveling,  Barometric  Leveling,  Topography,  Mining,  Sur- 
veying, the  Sextant,  and  other  Reflecting  Instruments,  Hydrograph- 
ical  Surveying,  and  Spherical  Surveying  or  Geodesy.  1  vol.,  Svo, 
173  pages. 

Elements  of  Plane  and  Spherical  Trigonom- 
etry, with  Applications.  By  Eugene  L.  Rich- 
AED8,  B.  A.,  Assistant  Professor  of  Mathematics  in  Yale  College. 
12mo.     295  pages. 

The  author  has  aimed  to  make  the  subject  of  Trigonometry  plain  to  begin- 
ners, and  much  space,  therefore,  is  devoted  to  elementary  definitions  and  their  ap- 
plications. A  free  use  of  diagrams  is  made  to  convey  to  the  student  a  clear  idea 
of  relations  of  magnitudes,  and  all  difficult  points  are  fully  explained  and  illustrated. 

Williamson's  Integral  Calculus,  containing  Appli- 
cations to  Plane  Curves  and  Surfaces,  with  numerous  Examples. 
12mo.     375  pages. 

Williamson's   Differential  Calculus,  containing 

the  Theory  of  Plane  Curves,  with  numerous  Examples.  12mo.  416 
pages. 

Perkins's  Elements  of  Algebra.   i2mo.  244  pages. 

Inventional  Geometry.     Science  Primer  Series.     18mo. 

The  Universal  Metric  System.    By  Alfred  Colin, 

C.  E.     12mo. 

D.  APPLETON  &  CO.,  Publishers, 

NEW  YORK,  BOSTON,  CHICAGO,  SAN  FRANCISCO. 


ASTRONOMY  AND  GEOLOGY. 


Lockyer's  Elements  of  Astronomy.  Accom- 
panied with  numerous  Illustrations,  a  Colored  Kepresentation  of  the 
Solar,  Stellar,  and  Nebular  Spectra,  and  Arago's  Celestial  Charts  of 
the  Northern  and  Southern  Hemisphere.  American  edition,  revised, 
enlarged,  and  specially  adapted  to  the  wants  of  American  schools. 
12mo.     312  pages. 

The  author's  aim  throughout  the  book  has  been  to  give  a  connected  view  of  the 
whole  subject  rather  than  to  discuss  any  particular  parts  of  it,  and  to  supply  facts 
and  ideas  founded  thereon,  to  serve  as  a  basis  for  subsequent  study.  The  arrange- 
ment adopted  is  new.  The  Sun's  true  place  in  the  Cosmos  is  shown,  and  the  real 
movements  of  the  heavenly  bodies  are  carefully  distinguished  from  their  apparent 
movements,  which  greatly  aids  in  imparting  a  correct  idea  of  the  celestial  sphere. 

The  fine  STAR-MAPS  OF  ARAGO,  showing  the  boundaries  of  the  constella- 
tions and  the  principal  stars  they  contain,  are  appended  to  the  volume. 

Science  Primer  of  Astronomy.    i8mo. 

Elements  of  Astronomy.  By  Robert  S.  Ball,  Pro- 
fessor of  Astronomy  in  the  University  of  Dublin,  and  Eoyal  Astron- 
omer of  Ireland.     12mo.    459  pages. 

Le  Conte's  Elements  of  Geolog^y.    A  Text-Book 

for  Colleges  and  for  the  General  Reader.  Revised  and  enlarged 
edition.    8vo.     633  pages. 

This  work  is  now  the  standard  text-book  in  most  of  the  leading  colleges  and 
higher-grade  schools  of  the  country.  The  author  has  just  made  a  thorough  re- 
vision of  the  work,  so  as  to  embrace  the  results  of  all  the  latest  researches  in 
geological  science. 

Nicholson's  Text-Book  of  Geology.    For  Schools 

and  Academies.     12mo.    277  pages. 

This  book  presents  the  leading  principles  and  facts  of  Geological  Science  in  as 
brief  a  compass  as  is  compatible  with  the  utmost  clearness  and  accuracy. 

Lyell's   Principles  of  Geology;  or,  The  Modern 

Changes  of  the  Earth  and  its  Inhabitants  considered  as  illustrative 
of  Geology.  Illustrated  with  Maps,  Plates,  and  Woodcuts.  2  vols., 
royal  8vo. 

Sir  Charles  Lyell  was  one  of  the  greatest  geologists  of  our  age.  In  this  work 
are  embodied  such  results  of  his  observation  and  research  as  bear  on  the  modem 
changes  in  the  earth's  structure  and  the  organic  and  inorganic  kingdoms  of  Nature. 

Science  Primer  of  Geology.    i8mo. 

D.  APPLETON  &  CO.,  Publishers, 

NEW  YORK,  BOSTON,  CHICAGO,  SAN  FRANCISCO. 


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